high quality rendering using ray tracing and photon mapping

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High Quality Rendering

using

Ray Tracing and Photon Mapping

Siggraph 2007 Course 8

Monday, August 5, 2007

Lecturers

Henrik Wann Jensen

University of California, San Diego

and

Per ChristensenPixar Animation Studios



Abstract

Ray tracing and photon mapping provide a practical way of efficiently

simulating global illumination including interreflections, caustics, color

bleeding, participating media and subsurface scattering in scenes with

complicated geometry and advanced material models.

This halfday course will provide the insight necessary to efficiently

implement and use ray tracing and photon mapping to simulate global

illumination in complex scenes. The presentation will cover the fun-

damentals of ray tracing and photon mapping including efficient tech-niques and data-structures for managing large numbers of rays and

photons. In addition, we will describe how to integrate the informa-

tion from the photon maps in shading algorithms to render global illu-

mination effects such as caustics, color bleeding, participating media,

subsurface scattering, and motion blur. Finally, we will describe re-

cent advances for dealing with highly complex movie scenes as well

as recent work on realtime ray tracing and photon mapping.



About the Lecturers

Per H. Christensen

Pixar Animation Studios

[email protected]

Per Christensen is a senior software developer in Pixar’s RenderMan group. His

main research interest is efficient ray tracing and global illumination in very com-

plex scenes. He received an M.Sc. degree in electrical engineering from the Tech-

nical University of Denmark and a Ph.D. in computer science from the University

of Washington in Seattle. His movie credits include ”Final Fantasy”, ”FindingNemo”, ”The Incredibles”, and ”Cars”.

Henrik Wann Jensen

University of California, San Diego

[email protected]

http://graphics.ucsd.edu/˜henrik

Henrik Wann Jensen is an Associate Professor at UC San Diego where he teaches

computer graphics. His main research interest is in the area of global illumina-

tion and appearance modeling. He received his M.Sc. degree and his PhD degree

from the Technical University of Denmark. He is the author of ”Realistic Image

Synthesis using Photon Mapping,” AK Peters 2001.



Contents

1 Introduction 9

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 What is photon mapping? . . . . . . . . . . . . . . . . . . . . . . 10

1.3 More information . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 11

Part 1: Per Christensen 12

2 Ray-tracing fundamentals 13

2.1 Historical background: ray tracing in the pre-computer age . . . . 14

2.2 Visibility ray tracing . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Ray intersection calculations . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Quadrilaterals (bilinear patches) . . . . . . . . . . . . . . 18

2.3.3 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.4 Implicit surfaces . . . . . . . . . . . . . . . . . . . . . . 21

2.3.5 NURBS surfaces . . . . . . . . . . . . . . . . . . . . . . 21

2.3.6 Subdivision surfaces . . . . . . . . . . . . . . . . . . . . 21

2.3.7 Displacement-mapped surfaces . . . . . . . . . . . . . . 22

2.3.8 Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Reflection models . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Shadow ray tracing . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Recursive ray tracing . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.1 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.2 Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Monte Carlo ray tracing . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.1 Distribution ray tracing . . . . . . . . . . . . . . . . . . . 29

5



2.7.2 Path tracing . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7.3 Soft shadows . . . . . . . . . . . . . . . . . . . . . . . . 302.7.4 Ambient occlusion . . . . . . . . . . . . . . . . . . . . . 30

2.7.5 Glossy reflections . . . . . . . . . . . . . . . . . . . . . . 30

2.7.6 Diffuse reflections . . . . . . . . . . . . . . . . . . . . . 31

2.7.7 Depth of field . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7.8 Motion blur . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Spatial acceleration data structures . . . . . . . . . . . . . . . . . 33

2.8.1 Bounding volume hierarchy . . . . . . . . . . . . . . . . 34

2.8.2 Which acceleration data structure is best? . . . . . . . . . 34

2.9 Ray differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.9.1 Ray propagation and specular reflection . . . . . . . . . . 352.9.2 Glossy and diffuse reflection . . . . . . . . . . . . . . . . 36

2.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Ray tracing in complex scenes 39

3.1 Many light sources . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Too many textures . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Multiresolution textures . . . . . . . . . . . . . . . . . . 40

3.2.2 Texture tiling . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.3 Multiresolution texture tile cache . . . . . . . . . . . . . 41

3.3 Geometric complexity . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.1 Instancing . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 Ray reordering and shading caching . . . . . . . . . . . . 42

3.3.3 Geometric stand-ins . . . . . . . . . . . . . . . . . . . . 42

3.3.4 Multiresolution tessellation . . . . . . . . . . . . . . . . . 43

3.4 Parallel execution . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 SIMD instructions . . . . . . . . . . . . . . . . . . . . . 44

3.4.2 Multiprocessors . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.3 Clusters of PCs . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Ray tracing in Pixar movies . . . . . . . . . . . . . . . . . . . . . 45

Part 2: Henrik Wann Jensen 47

4 A Practical Guide to Global Illumination using Photon Mapping 49

4.1 Photon tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 Photon emission . . . . . . . . . . . . . . . . . . . . . . 49



4.1.2 Photon tracing . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.3 Photon storing . . . . . . . . . . . . . . . . . . . . . . . 564.1.4 Extension to participating media . . . . . . . . . . . . . . 58

4.1.5 Three photon maps . . . . . . . . . . . . . . . . . . . . . 60

4.2 Preparing the photon map for rendering . . . . . . . . . . . . . . 60

4.2.1 The balanced kd-tree . . . . . . . . . . . . . . . . . . . . 61

4.2.2 Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 The radiance estimate . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.1 Radiance estimate at a surface . . . . . . . . . . . . . . . 63

4.3.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.3 The radiance estimate in a participating medium . . . . . 68

4.3.4 Locating the nearest photons . . . . . . . . . . . . . . . . 684.4 Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.1 Direct illumination . . . . . . . . . . . . . . . . . . . . . 73

4.4.2 Specular and glossy reflection . . . . . . . . . . . . . . . 74

4.4.3 Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4.4 Multiple diffuse reflections . . . . . . . . . . . . . . . . . 76

4.4.5 Participating media . . . . . . . . . . . . . . . . . . . . . 77

4.4.6 Why distribution ray tracing? . . . . . . . . . . . . . . . 77

4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.5.1 The Cornell box . . . . . . . . . . . . . . . . . . . . . . 79

4.5.2 Cornell box with water . . . . . . . . . . . . . . . . . . . 854.5.3 Fractal Cornell box . . . . . . . . . . . . . . . . . . . . . 86

4.5.4 Cornell box with multiple lights . . . . . . . . . . . . . . 87

4.5.5 Cornell box with smoke . . . . . . . . . . . . . . . . . . 88

4.5.6 Cognac glass . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5.7 Prism with dispersion . . . . . . . . . . . . . . . . . . . . 90

4.5.8 Subsurface scattering . . . . . . . . . . . . . . . . . . . . 91

4.6 Where to get programs with photon maps . . . . . . . . . . . . . 92

Part 3: Per Christensen 94

5 Photon mapping for complex scenes 95

5.1 Photon emission from complex light sources . . . . . . . . . . . . 95

5.2 Photon scattering from complex surfaces . . . . . . . . . . . . . . 97

5.3 The radiosity map . . . . . . . . . . . . . . . . . . . . . . . . . . 99



5.4 The radiosity atlas for large scenes . . . . . . . . . . . . . . . . . 100

5.4.1 Photon emission and photon tracing . . . . . . . . . . . . 1025.4.2 Radiosity estimation . . . . . . . . . . . . . . . . . . . . 102

5.4.3 Generating radiosity brick maps . . . . . . . . . . . . . . 102

5.4.4 Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5 Importance for photon tracing . . . . . . . . . . . . . . . . . . . 105

5.5.1 Importance . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.5.2 Importance emission and estimation . . . . . . . . . . . . 106

5.5.3 Photon tracing . . . . . . . . . . . . . . . . . . . . . . . 106

8



Chapter 1

Introduction

This course material describes in detail the practical aspects of the ray tracing and

photon map algorithms. The text is based on published papers as well as industry

experience. After reading this course material, it should be relatively straightfor-

ward to add an efficient implementation of the photon map algorithm to any ray

tracer and to understand the details necessary to make ray tracing and photon map-

ping robust in complex scenes.

1.1 Motivation

The photon mapping method is an extension of ray tracing. In 1989, Andrew

Glassner wrote about ray tracing [25]:

“Today ray tracing is one of the most popular and powerful tech-

niques in the image synthesis repertoire: it is simple, elegant, and eas-

ily implemented. [However] there are some aspects of the real world

that ray tracing doesn’t handle very well (or at all!) as of this writ-

ing. Perhaps the most important omissions are diffuse inter-reflections

(e.g. the ‘bleeding’ of colored light from a dull red file cabinet onto a

white carpet, giving the carpet a pink tint), and caustics (focused light,

like the shimmering waves at the bottom of a swimming pool).”

At the time of the development of the photon map algorithm in 1993, these prob-

lems were still not addressed efficiently by any ray tracing algorithm. The pho-

ton map method offers a solution to both problems. Diffuse interreflections and

caustics are both indirect illumination of diffuse surfaces; with the photon map

9



method, such illumination is estimated using precomputed photon maps. Extend-

ing ray tracing with photon maps yields a method capable of efficiently simulatingall types of direct and indirect illumination. Furthermore, the photon map method

can handle participating media and it is fairly simple to parallelize.

1.2 What is photon mapping?

The photon map algorithm was developed in 1993–1994 and the first papers on the

method were published in 1995. It is a versatile algorithm capable of simulating

global illumination including caustics, diffuse interreflections, and participating

media in complex scenes. It provides the same flexibility as general Monte Carlo

ray tracing methods using only a fraction of the computation time.The global illumination algorithm based on photon maps is a two-pass method.

The first pass builds the photon map by emitting photons from the light sources into

the scene and storing them in a photon map when they hit non-specular objects.

The second pass, the rendering pass, uses statistical techniques on the photon map

to extract information about incoming flux and reflected radiance at any point in

the scene. The photon map is decoupled from the geometric representation of the

scene. This is a key feature of the algorithm, making it capable of simulating

global illumination in complex scenes containing millions of triangles, instanced

geometry, and complex procedurally defined objects.

Compared with finite element radiosity, photon maps have the advantage thatno meshing is required. The radiosity algorithm is faster for simple diffuse scenes

but as the complexity of the scene increases, photon maps tend to scale better. Also

the photon map method handles non-diffuse surfaces and caustics.

Monte Carlo ray tracing methods such as path tracing, bidirectional path trac-

ing, and Metropolis can simulate all global illumination effects in complex scenes

with very little memory overhead. The main benefit of the photon map compared

with these methods is efficiency, and the price paid is the extra memory used to

store the photons. For most scenes the photon map algorithm is significantly faster,

and the result looks better since the error in the photon map method is of low fre-

quency which is less noticeable than the high frequency noise of general Monte

Carlo methods.

Another big advantage of photon maps (from a commercial point of view) is

that there is no patent on the method; anyone can add photon maps to their renderer.

As a result several commercial systems use photon maps for rendering caustics and

10



global illumination.

1.3 More information

For more information about photon mapping, all the practical details, the theory

and the insight for understanding the technique see:

Henrik Wann Jensen

Realistic Image Synthesis using Photon Mapping

AK Peters, 2001

This book also contains additional information about participating media and sub-

surface scattering. Finally, it contains an implementation with source code in C++of the photon map data structure.

1.4 Acknowledgements

Henrik would like to thank all the people he has worked with over the years on the

research that is contained within the course notes.

Per would like to thank his colleagues at Pixar and Pixar’s RenderMan team

for providing an inspiring environment and for their help and support. David Laur

and Julian Fong implemented large parts of the ray-tracing functionality in Pixar’s

RenderMan, which was used to generate many of the images in these course notes.Thanks to Wayne Wooten for digging out the data for the Pixar test scenes.

All images from Monsters, Inc., Cars, and Ratatouille are copyright c Disney

Enterprises, Inc. and Pixar Animation Studios.

11



12



Chapter 2

Ray-tracing fundamentals

This chapter provides an overview of the fundamental aspects of the ray tracing

algorithm. Ray tracing has its roots in the renaissance (or perhaps even earlier),

and was used for drawing images with correct perspective foreshortening and for

design of mirrors and lenses for telescopes and other optical instruments.

Ray tracing can be used for visibility testing (projecting the 3D scene to a

2D image) and shadows, but it really comes into its right element when used to

compute specular reflections and refractions. The basic ray tracing algorithm is

very simple and elegant; nevertheless it can compute visual phenomena that are

very hard to compute with any other method. Monte Carlo ray tracing adds further

effects such as soft shadows and ambient occlusion, glossy and diffuse reflections,

depth-of-field, and motion blur.

This chapter starts with a brief history of ray tracing, and then shows how to

use ray tracing for direct rendering. It then provides an overview of ray intersec-

tion calculation methods for various types of surfaces, a key part of ray tracing.

Then follows a brief overview of reflection models. With these basics covered,

the use of ray tracing to render shadows, reflection and refraction, soft shadows,

glossy and diffuse reflections, depth-of-field, and motion blur is described. Vari-

ous spatial acceleration data structures are used to speed up ray tracing in scenes

with many objects; those are only described very superficially. At the end there is

a description of ray differentials, a relatively new concept that is useful for many

applications including texture filtering and tessellation. The last section contains

recommendations for further reading.

13



2.1 Historical background: ray tracing in the pre-computer

age

Ray tracing was used long before the electronic computer was invented. Figure 2.1

shows ray tracing in the year 1525 — back then, the “computer” was the artist’s

assistant and rays were strands of thread. Albrecht Durer (1471–1528), a German

renaissance painter and engraver, used this device to render images with correct

perspective projection [22]. Points on the object (a lute) are projected onto the

image. Nowadays we would call this process a “projection of 3D points onto a 2D

image”.

Figure 2.1: Mechanical creation of a perspective image (Durer, 1525).

Ray tracing was also used for lens design for microscopes, telescopes, binocu-

lars, and cameras. Sir Isaac Newton (1642–1727) showed reflection and refraction

of rays in his famous 1704 book Opticks [50]. One of his illustrations is shown in

figure 2.2.

2.2 Visibility ray tracing

Rendering consists of computing the color of each point in an image. This is done

by projecting the 3D scene onto a 2D image. Ray tracing is one of the most popular

rendering methods. The basic ray tracing algorithm consists of two main calcula-

tions per pixel: find the nearest surface point and compute the color at that point.

It determines the visibility of object surfaces by following imaginary rays from the

14



Figure 2.2: Prism with refracted rays (Newton, 1704).

viewer’s eye to the objects in the scene. This limited type of ray tracing is some-

times referred to as ray casting. Appel [3] was the first to use a computer to renderray-traced images.

The simplest ray tracing algorithm is as follows:

for each pixel do

compute ray for that pixel

for each object in scene do

if ray intersects object and intersection is nearest so far then

record intersection distance and object color

set pixel color to nearest object color (if any)

Computing the ray corresponding to a pixel is very simple: the ray origin is

at the viewpoint, and the ray direction is from the viewpoint to the pixel position

in the image plane. The computation of the intersection of a ray with an object is

fairly simple; we cover the details of this in the following section (2.3).

A simple way to improve the image quality is to shoot several rays per pixel.

This reduces aliasing effects such as “jaggies” and staircase effects along object

silhouettes.

Figure 2.3 shows a simple ray-traced image of two teapots. The image shows

the shape of the teapots, as seen from the viewpoint. However, the image is very

simplistic since the teapots are simply rendered black. To improve on this, we need

shading and reflection models which are described in section 2.4.

15



Figure 2.3: Simple image of two teapots.

2.3 Ray intersection calculations

At the heart of all ray tracing algorithms is the computation of ray-object intersec-

tions. If there is more than one intersection we usually want the nearest. (However,

for shadow rays we often just need to know whether there is any intersection; it

doesn’t matter if it’s the nearest one.) This section provides an overview of ray-

object intersection calculation methods.

First a definition: a ray is a semi-infinite line. It is defined by an origin o and a

direction d: p(t) = o + t d for all t ≥ 0.

2.3.1 Triangles

A triangle abc is defined by its three vertices a, b, c. The normal of the triangle

can be computed (on-the-fly or just once and stored with the triangle data) using

the cross product of two of the triangle edges, for example ( b− a) and (c− a):

n = ( b− a)× (c− a) .

The ray-triangle intersection point p must be along the ray (i.e. p = o+ t d) and

must be in the plane of the triangle (i.e. the vector from a triangle vertex to p must

be perpendicular to the triangle normal), ( p−a) ·n = 0. From these two equationswe get:

0 = ( p− a) · n= (o + t d− a) · n= (o− a) · n + t d · n

16



Solving for t we get:

t = (a− o) · n d · n

.

If the dot product d · n is 0 the ray is parallel to the plane and there is no

intersection. If the computed t is negative there is an intersection but it is behind

the ray origin, so we reject it. We also reject the intersection if it is further away

than a previously found intersection for that ray.

Given the distance t, we can compute the ray-plane intersection point: p =

o + t d. Next we must check whether this point is inside or outside the triangle.

We do this by computing the barycentric coordinates (u, v) of the hit point. The

barycentric coordinates of p are defined by

a + u( b− a) + v(c− a) = p

— as illustrated in figure 2.4.

a b

c

u( b− a)v(c− a)

p

p

Figure 2.4: Triangle abc and barycentric coordinates (u, v) of point p.

Since the vectors in the equation above have three components (x, y, and z)

we have three equations to determine the two unknowns u and v, and we are free

to choose two of the three equations to solve. In practice, one should pick the

two equations where the floating point number precision is highest. Finally, when

the barycentric coordinates have been computed, we can determine whether the

intersection point is inside the triangle or not. If u

≥0 and v

≥0 and u + v

≤1

then the point is inside (or on the edge of) the triangle and we finally have an actual

intersection!

Further implementation details, optimizations, and pseudo-code can be found

in e.g. Glassner [25] and Moller and Trumbore [48].

17



2.3.2 Quadrilaterals (bilinear patches)

A quadrilateral (or bilinear patch) abcd is defined by four vertices a, b, c, d. We

can express all points p on the surface of the quadrilateral as a bilinear combination

of the four vertices:

(1 − u)(1 − v)a + u(1− v) b + (1 − u)vc + uv d = p .

Figure 2.5 shows a quadrilateral with iso-lines for u and v. The point p is on the

quadrilateral if both u and v are in the range [0, 1].

a b

c d

p p

u = 00.250.50.75u = 1v = 00.250.50.75v = 1

Figure 2.5: Quadrilateral abcd and bilinear coordinates (u, v) of point p.

Quadrilaterals are not flat in general, but for flat quadrilaterals we can compute

the intersection point more efficiently than for the general case. We will look at the

general, non-planar case first.

Non-planar quadrilaterals

A ray can intersect a non-planar quadrilateral at 0, 1, or 2 points.

Plugging the ray equation p = o + t d into the quadrilateral equation above

gives 3 equations (one for each of the x, y, and z coordinates) with 3 unknowns

t, u, and v. Unfortunately the equations are non-linear. However, a fairly effi-

cient method is to first solve a quadratic equation for u. If u is in the [0, 1] range

then compute the corresponding v. If v is also in [0, 1] then compute t. Reject the

intersection if t is negative or larger than the previous nearest hit distance. Imple-

mentation details for this method can be found in Stephenson [70]. The normal

at the hit point can be computed e.g. by bilinear combination of the four vertex

normals.

18



A more efficient method can be used if the quadrilateral is small. In that case

it is often sufficient to approximate the quadrilateral as two triangles. The inter-section points might be slightly incorrect, and the u and v iso-lines get a “kink”

along the diagonal as shown in figure 2.6. However, if the quadrilateral is smaller

than e.g. a pixel this is a very useful optimization, and it is used a lot in practice for

calculating ray intersections with finely tessellated geometry.

a b

c d

Figure 2.6: Quadrilateral abcd approximated by two triangles.

Planar quadrilaterals

A ray can intersect a planar quadrilateral at 0 or 1 points. One method computes

the ray intersection with the plane the quadrilateral is in (using the same calcula-

tion as for ray-triangle tests) and then determines whether the intersection point

is inside or outside the quadrilateral. An even more efficient method for convex

planar quadrilaterals has been presented by Lagae and Dutre [43].

If the quadrilateral is a rectangle or square it is even simpler to determine the

bilinear coordinates u and v — it can be done with just dot products (projecting p

onto two of the edges).

2.3.3 Quadrics

The class of quadric surfaces consists of disks, spheres, cylinders, cones, ellipsoids,

paraboloids, and hyperboloids.

Disks

A disk is defined by its center c, normal n, and radius r. The center and normal

of the disk define a plane. Finding a ray-disk intersection is very similar to ray-

triangle intersection testing. We first compute the ray-plane intersection point p,

19



and check that the distance t is positive and smaller than the previous nearest inter-

section. The intersection point is on the disk if ( p− c)2

≤ r2

.Disks are used a lot in practical rendering, for example for rendering of particle

systems.

Spheres

A sphere is defined by its center c and radius r. If there is an intersection, the

intersection point must be somewhere along the ray, and must be on the surface of

the sphere. To find the intersection point we plug the ray equation p = o + t d into

the sphere equation ( p− c)2 = r2:

0 = ( p− c)2

− r2

= p2 − 2( p · c) + c2 − r2

= (o + t d)2 − 2(o + t d) · c + c2 − r2

= o2 + 2t(o · d) + t2 d2 − 2(o · c)− 2t( d · c) + c2 − r2

= d2t2 + 2 d · (o− c)t + (o− c)2 − r2

This is a quadratic equation in t. The two solutions are

t1 =−B + D

2Aand t2 =

−B −D

2A

— with A = d2, B = 2 d · (o − c), C = (o − c)2 − r2, and the discriminant D

is D =√B2

−4AC . (If we know in advance that the ray direction is normalized

then A = 1.) If the discriminant D is negative there is no (real) solution and the ray

does not hit the sphere. If the discriminant is zero the ray is tangent to the sphere,

and there is only one intersection point. If the discriminant is positive there are

two intersection points; the nearest intersection point is the one with the smallest

non-negative value of t. Given the intersection distance t we can then compute the

intersection point p.

The normal at the intersection point is n = p− c.

Other quadrics

Ray intersections with cylinders, cones, ellipsoids, paraboloids, and hyperboloids

can be computed by solving a quadratic equation in a similar fashion, see e.g. Glass-

ner [25].

Alternatively, ellipsoids can also be tested by transforming the ray by the same

transformation that transforms the ellipsoid to a sphere; then the intersection can

20



be computed as a ray-sphere intersection. (If there is a hit, the hit point and normal

must be transformed back.)

2.3.4 Implicit surfaces

An implicit surface is defined by a function f : the surface is the set of points p

where the value of the function is 0, f ( p) = 0. So to find the ray-surface intersec-

tion we have to determine the (nearest) point p along the ray where f ( p) is 0:

f (o + t d) = 0

This can be done using e.g. Newton-Raphson iteration or other iterative methods.

An efficient algorithm is described by Sherstyuk [62]. The surface normal at the

intersection point is given by the gradient of the function at that point:

n = f ( p) =

∂f ( p)

∂x,∂f ( p)

∂y,∂f ( p)

∂z

.

2.3.5 NURBS surfaces

NURBS surfaces [23, 58] are widely used in the CAD industry for modeling cars,

airplanes, etc. NURBS surfaces can be intersection tested directly (see Kajiya [36],

Martin et al. [47], and Abert et al. [1]) or tessellated into polygon (triangle or

quadrilateral) meshes and then intersection tested.

NURBS surfaces often have trimming curves, indicating parts of the surface

that are “cut away”. In this case, the ray-NURBS intersection point must be

checked against the trimming curve to determine whether the intersection point

is inside or outside the trim curves. If the hit point is outside the trimming curve it

should be rejected.

2.3.6 Subdivision surfaces

Subdivision surfaces [10, 21] are widely used in the movie industry to model

smooth surfaces with complex topology, for example humans, animals, etc. [20].

The most common subdivision surface types are Catmull-Clark [10] and Loop [46].

Direct ray tracing of subdivision surfaces is somewhat tricky, particularly near ex-

traordinary vertices; see Kobbelt et al. [41] and Pharr and Humpheys [56] for al-

gorithms. Another strategy is to tessellate the subdivision surface into a polygon

mesh and then ray trace the mesh.

21



2.3.7 Displacement-mapped surfaces

Displacement maps are used to alter the shape of surfaces, for example to add

details such as wrinkles, dents, large bumps, scratches, reptile scales, etc.

A method by Smits et al. [68] computes the displacements “on the fly” to de-

termine the ray intersection points. The advantage of their method is that no extra

memory is required. The disadvantage is that the displacements have to be along

the normals and that the displacement function is evaluated repeatedly (which can

be very time-consiming). It is faster and more general (but also more memory con-

suming) to tessellate larger patches of the surfaces, displace the tessellated points,

and store the displaced points in a cache [55, 17].

2.3.8 Boxes

As we shall see later, bounding boxes are extremely useful for speeding up ray

tracing of complex scenes.

A general box is defined by a vertex and three vectors, see figure 2.7(a). A

straightforward intersection test would test each of the six faces for intersection.

A faster test can be implemented by utilizing the fact that only the three faces that

face toward the ray origin need to be tested.

(a) (b) p

p pmin

p pmax

Figure 2.7: Boxes: (a) box with general orientation; (b) an axis-aligned box.

Axis-aligned boxes can be intersection-tested even more efficiently. An axis-

aligned box consists of two rectangles in each of the xy, xz, and yz planes. An

axis-aligned box is defined by its minimum and maximum vertices pmin and pmax

— as illustrated in figure 2.7(b). We can consider the box the intersection of three

infinite slabs of space. Smits [67] describes a very efficient ray intersection test

that utilizes IEEE floating-point conventions to deal gracefully and efficiently with

divisions by 0, thereby streamlining the code.

If the boxes are used as bounding boxes, we don’t need to know the nearest

22



intersection point and normal, we just need to know whether the ray intersects the

box or not.

2.4 Reflection models

There are many reflection models that can be used with ray tracing, but since re-

flection models are not a priority for this course, we will only give a very brief

overview here.

Diffuse reflection can be modeled with Lambert’s cosine law: the reflected

light is proportional to the cosine of the angle between the incident light and the

surface normal [44]. A more accurate model of diffuse reflection was developed

by Oren and Nayar [51]. Glossy and specular surfaces have highlights. The high-

lights can be computed with the Phong cosine-power formula [8, 6], with Ward’s

isotropic and anisotropic reflection models [81], or with a number of other reflec-

tion models [24]. Surfaces can also have textures assigned to them to modulate the

reflection parameters [9, 32]. The textures can be 2D images or 3D tables, or can

be computed procedurally.

The illumination of the surfaces is provided by light sources. The simplest

light source types are point lights, spot lights, and directional lights. In contrast,

very complex light sources [5] are used in movie production. Such light sources

can project images like a slide projector and can have complex intensity fall-off,

barn doors, cucoloris (“cookies”), etc. In addition, it is very common to add an

“ambient” term to the illumination, i.e. a constant amount of illumination — inde-

pendent of surface position and orientation. The ambient light “fills in” the color

on surfaces facing away from the light sources (otherwise those surfaces would be

completely black). The ambient term is a cheap hack to approximate the light that

bounces around in a real scene.

In this chapter we will only use very simple reflection models: Lambert for

diffuse reflection and Phong for specular reflection. We’ll use a point light source

and ambient for illumination. Figure 2.8 shows two teapots on a square. The sur-

faces are Lambert diffuse with Phong specular highlights. In addition, the square

has a checkerboard texture map. The scene is illuminated by a point light and a

dim ambient light.

23



Figure 2.8: Teapots and a square showing Lambert diffuse reflection, Phong spec-

ular highlights, a checkerboard texture map, and simple illumination.

2.5 Shadow ray tracing

So far, we have only used ray tracing to determine which objects are visible in

which image pixels. The first additional use of ray tracing is for shadow computa-

tion: we can determine whether a point is in shadow by tracing a ray from the point

to the light source [3]. If the ray hits an opaque object along the way, the object is

in shadow; if not, it is illuminated. When computing ray-object intersections for

opaque shadows, we only care about hit or no hit; not the intersection point and

normal. Figure 2.9(a) shows a few examples of shadow rays.

For point lights and spot lights we trace rays between the surface points and

the light source position. For directional light sources, we trace parallel rays from

the surface points in the direction of the light. Figure 2.9(b) shows shadows from

a point light in the familiar teapot scene.

(a) (b)

Figure 2.9: (a) Shadow rays. (b) Teapots with ray-traced shadows.

24



If the objects are opaque, any hit will suffice to determine shadow. But if the

objects are semitransparent (as e.g. stained glass), we need to get the transmissioncolor of all the intersected surfaces between the point and light source, and then

composite the transmission colors by multiplying each color component.

2.6 Recursive ray tracing

When a ray hits a surface with specular reflection or refraction, computing the

color there may require tracing more rays — called reflection rays and refraction

rays, respectively. Those rays may hit other specular surfaces, causing more rays

to be traced, and so on. Hence the term recursive ray tracing. Figure 2.10 shows

a recursive “tree” of reflection rays. This technique is also known as classical ray

tracing or Whitted-style ray tracing since it was introduced by Turner Whitted in

1980 [83].

Figure 2.10: Reflection and refraction ray tree.

2.6.1 Reflection

We shoot reflection rays to compute reflection from specular surfaces such as shiny

metals. To compute the reflection direction we use the law of reflection: the angle

of reflection equals the angle of incidence. Furthermore, the reflection direction is

constrained to be in the plane spanned by the incident direction and the surface nor-

mal. Figure 2.11 shows the incident direction i, the surface normal n, the reflection

direction r, as well as the angles of incidence and reflection (θi and θr).

25



in

r

i

d 2 dθiθr

Figure 2.11: Reflection direction geometry.

As can be seen in the figure, r = i + 2 d, where d is the projection of − i onto

the normal n and can be computed as

d = (− i · n)nn2 .

With this, the reflection direction r is

r = i + 2 d = i− 2( i · n)n

n2.

This formula does not require the vectors to be normalized. However, if we know

that the normal n is normalized we can avoid the division by n2.

Figure 2.12 shows two chrome teapots with ray-traced reflections. The close-

up shows how beautifully distorted such reflections can be, even on relatively sim-

ple geometry like this.

(a) (b)

Figure 2.12: (a) Chrome teapots with reflections. (b) Close-up showing the reflec-tions more clearly.

In this image, the reflection amount was set to 100%. To be more physically

correct, one can scale the reflection amount depending on the reflection angle. The

26



amount of reflection can be computed using the Fresnel formulas [24, 26] or using

a fast approximation by Schlick [61].

2.6.2 Refraction

Dielectric materials such as water and glass exhibit both reflection and refraction.

The refraction direction depends on the two materials’ index of refraction η.

The index of refraction of vacuum is 1, for air it is 1.0003, for water it is 1.33, for

glass it is in the range 1.5–1.75, and for diamond it is 2.42.

Figure 2.13 shows the incident direction i, surface normal n, the refraction

(transmission) direction t, and the angles of incidence and refraction (θi and θt).

ηi is the index of refraction of the material of the incident ray, and ηt is the index

of refraction of the transmitting material.

in

−n tθiθt

ηiηt

Figure 2.13: Refraction direction geometry.

The direction of the refraction ray can be computed using the law of refraction

(often called Snell’s law or Descartes’ law although it was possibly known much

earlier by Ibn Sahl [33]). The relationship between the incident and transmitted

direction is

ηi sin θi = ηt sin θt .

From this we can derive that the transmitted direction is

t = −η i +η cos θi −

1− η2(1− cos2 θi)

n

— for η = ηi/ηt and normalized i and n. (For a derivation please see e.g. Glass-

ner [25] or Shirley and Morley [64].)

If the quantity under the square root is negative there is no refraction. This is

called total internal reflection and can only occur when the light transfers from a

material with high index of refraction to a material with lower index of refraction.

27



The angle where the quantity under the square root is zero is called the critical

angle.The amount of reflection and refraction depends on the incident angle and in-

dex of refraction. The amount can be computed with the Fresnel formulas; these

formulas are omitted here, but can be found in e.g. the books by Foley et al. [24]

and Glassner [26]. A fast and very useful approximation to the Fresnel formulas

was introduced by Schlick [61].

Figure 2.14 shows two glass teapots with ray-traced reflections and refractions.

Notice how the glass bends the light that is refracted through it. The amount of

reflection and refraction in this image is determined by Fresnel’s formulas.

Figure 2.14: Glass teapots with reflection and refraction.

An effect that is often added to render glass and water is distance attenuation

(exponential intensity fall-off according to Beer’s law). This is simple to implement

and is described in e.g. Shirley and Morley’s book [64].

2.7 Monte Carlo ray tracing

In the previous section, all ray directions were determined deterministically. In this

section we’ll look at effects that are computed with Monte Carlo ray tracing (also

known as stochastic ray tracing), i.e. ray tracing where the ray origins, directions,

and/or times are computed using random numbers. Monte Carlo ray tracing is often

divided into two categories: distribution ray tracing and path tracing.

28



2.7.1 Distribution ray tracing

Distribution ray tracing [19] shoots multiple rays from each surface point to sample

area lights, glossy and diffuse reflection, and many other effects. Figure 2.15 shows

a tree of reflection and refraction rays for distribution ray tracing. As the figure

shows, distribution ray tracing is prone to an explosion in the number of rays after

a few levels of reflection; to avoid this it is common to reduce the number of rays

after a few levels of reflection. With distribution ray tracing, it is quite easy to

ensure a good distribution of ray directions at reflection points, for example by

stratifying the directions.

Figure 2.15: Reflection and refraction tree for distribution ray tracing.

2.7.2 Path tracing

Path tracing [37] is a variation of distribution ray tracing where only a single re-

flection and refraction ray is shot for each point. This avoids the explosion in the

number of rays, but a simple implementation would lead to very noisy images. To

compensate for that, many visibility rays are traced through each pixel. An ad-

vantage of path tracing is that since many visibility rays are shot per pixel, camera

effects like depth-of-field and motion blur can be incorporated at little extra cost.

On the other hand, it is harder to ensure a good distribution of reflection rays (for

example through stratification) than for distribution ray tracing.

Put succinctly, distribution ray tracing shoots most rays deeper in the ray tree,

while path tracing shoots most visibility rays.

29



2.7.3 Soft shadows

Area light sources cause soft shadows. (The region in between complete shadow

and complete illumination is called the penumbra.) Soft shadows can be computed

by shooting shadow rays to random points on the surface of the area light source.

Figure 2.16(a) shows shadow rays from three surface points to a triangular area

light source; some of the rays hit an object. Figure 2.16(b) shows soft shadows in

the familiar teapot scene. In this image, the light source is spherical and the soft

shadow is computed with distribution ray tracing.

(a) (b)

Figure 2.16: (a) Shadow rays to an area light source. (b) Teapots with soft shadows.

2.7.4 Ambient occlusion

Ambient occlusion [85, 45] can be thought of as illumination by an extremely large

area light source, namely the entire hemisphere above each point. this is similar to

the illumination outside on an overcast day.

Figure 2.17(a) shows ambient occlusion rays from two surface points. At the

left point most of the rays hit an object, so the occlusion is high; at the right point

few rays hit an object, so there is little occlusion. Figure 2.17(b) shows ambient

occlusion in the teapot scene. This figure shows pure ambient occlusion; this can

of course be combined with surface colors, textures, etc.

2.7.5 Glossy reflections

Glossy reflection of indirect light can be computed by shooting rays within the

directions of the glossy reflection distribution. For a given incident direction and

a pair of random numbers, the reflection model provides a reflection direction.

30



(a) (b)

Figure 2.17: (a) Ambient occlusion rays. (b) Teapots with ambient occlusion.

Figure 2.18 shows glossy reflections in the two teapots; reflections are computed

using Ward’s (isotropic) glossy reflection model [81]. (There is also an anisotropic

version of this shading model — it can be used to render glossy reflections from

e.g. brushed metal surfaces.)

(a) (b)

Figure 2.18: (a) Teapots with glossy reflections of direct and indirect light.

(b) Close-up showing the glossy reflections more clearly.

Similarly, glossy refraction can be computed by distributing the rays around

the refraction direction. This gives the appearance of slightly frosted glass.

2.7.6 Diffuse reflections

Ward et al. [82] used wide distribution ray tracing to compute indirect diffuse light.

The distribution of reflection rays covers the entire hemisphere above each point,

with a cosine-weighted distribution such that more rays are traced in directions

toward the pole than near the equator. An example of this can be seen in figure 2.19.

31



In this image there is no ambient light source; any light in shadow regions is due

to diffuse reflection of indirect light. Note in particular how the white checkersare reflected in the bottom of the teapots, and how the spout on the right teapot

casts light onto the nearby part of the teapot body. This effect is often called color

bleeding (although in this case the “color” is white) and can tint surfaces with the

colors of nearby objects.

Figure 2.19: Teapots and square with diffuse reflection of direct and indirect light.

2.7.7 Depth of field

Real cameras built with lenses have a finite aperture opening. As a result, they

can only focus at a particular distance, and objects that are far from that distanceare blurry. (An exception is pin-hole cameras that have a nearly infinitely-small

aperture opening so all distances are in focus.) In computer graphics we can sim-

ulate the finite aperture opening by tracing rays with slightly varying origins and

directions as shown in figure 2.20(a). Figure 2.20(b) illustrates the depth-of-field

effect: the front teapot is (mostly) in focus while the rear teapot is out of focus.

Distribution ray tracing for more realistic camera models is described by Kolb et

al. [42].

2.7.8 Motion blur

In real cameras the shutter has to be open a finite amount of time to capture enough

light on the film or CCD chip. If an object is moving within the shutter opening

time, it will be blurry in the image. To render this effect, we can shoot the rays at

different times within the shutter interval. When intersection testing we move the

32



(a) (b)

Figure 2.20: (a) Finite aperture. (b) Teapots with depth of field.

objects to the appropriate time for the ray.Figure 2.21 shows two motion blurred teapots. The chrome teapot is moving

while the diffuse teapot is both moving and rotating around its own axis. The

teapots themselves are blurred, and their reflections and shadows are also blurred.

Figure 2.21: Motion-blurred teapots.

2.8 Spatial acceleration data structures

For complex scenes, it would be hopelessly inefficient to test every object for in-

tersection with every ray. We therefore organize the objects in a hierarchy so that

a large fraction of the objects can be rejected quickly.

The most important characteristics of an acceleration data structure are con-

struction time, memory use, and ray traversal time. Depending on the application,

different emphasis may be put on each of these characteristics. For rendering of

33



sequences of images (for example for interactive visialization or for “shot” ren-

dering for movies) it is also desirable to choose an acceleration data structure thatcan be efficiently updated with incremental geometry changes, see for example

Reinhard et al. [60] and Wald et al. [76].

There is a bewildering array of acceleration data structures: bounding vol-

ume hierarchies, uniform grids, hierarchical grids, BSP-trees, kd-trees, octrees,

5D origin-direction trees, bounding interval hierarchies, and so on. Here we will

only describe one acceleration data structure, the bounding volume hierarchy, in

detail.

2.8.1 Bounding volume hierarchy

A bounding volume hierarchy (BVH) organizes the objects and their bounding

volumes into a tree [39]. The root of the tree is a bounding volume containing the

entire scene. The most commonly used bounding volume is the axis-aligned box

since such boxes are easy to compute and combine.

For example, the BVH for the teapot scene has five levels of bounding boxes.

The top level consists of a single bounding box for the entire scene. The next

levels contains the bounding boxes of the two teapots and the square. Each teapot

consists of four parts: body, lid, handle, and spout. Each part has a bounding box.

The teapot body consists of eight Bezier patches, each with its own bounding box.

For a tessellated Bezier patch, each group of quadrilaterals can have a bounding

box for efficient ray intersection testing.The scene modeling hierarchy can be used directly, as in the teapot scene ex-

ample. Another strategy is to split the geometry such that the surface areas are

approximately equal in each part [27]. Smits’ article [67] contains much good

advice on efficient construction and traversal of bounding volume hierarchies.

When a ray needs to be intersection-tested with the objects in the scene, the

first step is to check for intersection with the bounding box of the entire scene. If

the ray hits the bounding box, the bounding boxes of the children are tested, and

so on. When a leaf of the hierarchy is reached, the object represented by the leaf

must be intersection tested.

2.8.2 Which acceleration data structure is best?

None of these acceleration data structures is consistently faster than the other.

Which one is optimal for a given scene depends on the scene characteristics, and

34



whether the emphasis is on fast construction, fast updates, fast ray traversal, or

compact memory use. For a detailed analysis please refer to Havran’s Ph.D. the-sis [30] and the discussions on Ray Tracing News [29].

2.9 Ray differentials

Even though ray differentials are a fundamental property of rays, their use for ray

tracing is relatively new. They are useful for many applications including texture

filtering and tessellation, as will become clear from the following chapters. A ray

differential describes the differences between a ray and its — real or imaginary —

“neighbor” rays. The differentials give an indication of the beam size that each ray

represents, as illustrated in figure 2.22.

rayneighbor ray

neighbor rayray beam

Figure 2.22: Rays and ray beam.

2.9.1 Ray propagation and specular reflection

Igehy’s ray differential method [34] keeps track of ray differentials as rays are prop-

agated and specularly reflected and refracted. The curvature at surface intersection

points determines how the ray differentials and their associated beams change after

specular reflection and refraction. For example, if a ray hits a highly curved, con-

vex surface, the specularly reflected ray will have a large differential (representing

highly diverging neighbor rays).

Figure 2.23 shows ray-traced specular reflections. In the left image no ray

differentials are computed and the texture filter width is zero; hence the aliasing

artifacts. In the right image, ray differentials are used to determine the proper

texture filter size. To show the differences clearly, the resolution of the images is

very low (200×200 pixels), only a single reflection ray was shot per pixel, and

pixel filtering was turned off.

35



(a) (b)

Figure 2.23: Reflections: (a) without use of ray differentials; (b) with.

2.9.2 Glossy and diffuse reflection

Suykens and Willems [73] generalized ray differentials to glossy and diffuse reflec-

tions. For distribution ray tracing of diffuse reflection or ambient occlusion, the ray

differential corresponds to a fraction of the hemisphere. The more rays are traced

from the same point, the smaller the subtended hemisphere fraction becomes. If

the hemisphere fraction is very small, a curvature-dependent differential (as for

specular reflection) becomes dominant.

2.10 Further reading

For further information about ray tracing, the best starting point is one of the ex-

cellent books dedicated to ray tracing: An Introduction to Ray Tracing edited by

Glassner [25], and Realistic Ray Tracing by Shirley and Morley [64]. There are

also several good books about rendering in general that include ray tracing [24, 26,

63, 56].

Eric Haines has compiled the on-line Ray Tracing News [29] since 1987. It

contains lots of discussions about the finer points of ray tracing, new developments

and insights over the years, etc.

Furthermore, there have been several SIGGRAPH courses on different aspects

of ray tracing, for example the Monte Carlo Ray Tracing course in 2003 [35] and

the Interactive Ray Tracing course in 2006 [65].

A series of symposia dedicated to ray tracing was recently started. The first

Ray Tracing Symposium was in Salt Lake City in September 2006. It was a great

36



event, and the proceedings [77] are full of the latest ray tracing research results.

The next symposium will be in Ulm, Germany in September 2007. We expect itand the following symposia to be as inspiring as the first.

37



38



Chapter 3

Ray tracing in complex scenes

In the previous chapters we have mainly looked at simple scenes with few light

sources, relatively few simple objects, and very simplistic shading. However, in

practical applications such as movie production, the scenes are much more com-

plex:

• Thousands of light sources.

• More textures than can fit in memory.

• More geometry than can fit in memory (in tessellated form).

• Very complex, programmable shaders for displacement, illumination, and

reflection. (10,000s lines of code.)

In addition, the images have to be of very high quality:

• High resolution.

• Motion blur.

• Depth of field.

• No spatial or temporal aliasing (no staircase effects, “crawlies”, popping,etc.)

In this chapter we will describe various techniques to render high-quality im-

ages of such complex scenes.

39



3.1 Many light sources

The main expense in computing the direct illumination from a light source is typi-

cally computing the shadows. If shadow maps [59] are used, one has to render and

manage a shadow map for each light source. If ray tracing is used and if we had

to trace at least one shadow ray for each light source, the render times would be

unacceptably long.

Fortunately, shadows from many light sources can be dealt with by sorting

the light sources based on their potential illumination. Some light sources are so

distant and their illumination so dim that they can be approximated very coarsely.

At each surface point, the direct illumination of each light source is computed, then

the lights are sorted according to illumination strength, and finally a probabilistic

selection is done of which lights to compute shadows for, which ones to compute

without shadows, and which ones to skip. Details can be found in the articles by

Ward [80] and Shirley et al. [66] and in Shirley and Morley’s book [64].

3.2 Too many textures

When the textures required to render an image exceeds the available memory, it

becomes essential to read the textures from disk on demand, to read the textures

only at the required resolution, and to cache the textures in memory.

3.2.1 Multiresolution textures

A texture MIP map [84] is a hierarchy of representations of a texture. Each level is

a down-sampled version of the next finer level; typically a pixel at a given level is

the average of four pixels at the next finer level. Figure 3.1 shows the six coarsest

levels of a texture MIP map. The coarsest level consists of a single pixel, the next

level consists of four pixels, and so on.

· · ·

Figure 3.1: Texture MIP map.

According to Peachey [53], Hanrahan was the first to make the observation

40



that for directly visible geometry and fixed image resolution, the required number

of texture pixels is roughly constant: A wide-angle view of a given scene containsmany objects, but little texture detail is visible on each object. On the other hand,

a close-up view of one of the objects shows the texture of that object at a much

higher resolution, but does not show the other objects. All in all, the total number

of texture pixels seen in the two images is roughly the same if the appropriate MIP

map levels are chosen.

Fortunately this constant nature is true for recursive ray tracing as well — but

only if we use ray differentials to determine the appropriate texture resolution at

ray hit points.

3.2.2 Texture tiling

It is advantageous to tile the textures so that groups of nearby pixels are read from

disk to memory together. Figure 3.2 shows three levels of a tiled texture MIP map.

In this example, each tile contains 16×16 pixels. The coarsest MIP map levels

(levels 0–3) can be squeezed into a single tile (not shown here). MIP map level 4

consists of a single tile. The next level has 2×2 tiles (still with 16×16 pixels in

each tile). The next level has 4×4 tiles, and so on.

· · ·

Figure 3.2: Tiled texture MIP map.

3.2.3 Multiresolution texture tile cache

Peachey [53] introduced a multiresolution texture tile caching scheme. He found

that texture accesses are highly coherent for rendering of directly visible geometry,

and that a cache size of 1% of the total texture size is sufficient. We have observed a

similar result for ray tracing when ray differentials are used to select the appropriate

MIP map level for texture lookups [17]. We choose the level where the texture

pixels are approximately the same size as the ray beam cross-section. Incoherent

41



rays have wide ray beams, so coarse MIP map levels will be chosen. The finer MIP

map levels will only be accessed by rays with narrow ray beams; fortunately thoserays are coherent so the resulting texture cache lookups will be coherent as well.

3.3 Geometric complexity

This section discusses methods for rendering large, complex scenes on a single PC.

Utilizing clusters of PCs for parallel speed-ups and even larger scenes is described

in section 3.4.

3.3.1 Instancing

Instancing can sometimes be a great way to generate geometric complexity in ascene. For example, it is simple to render a field full of sunflowers: only a few

unique sunflower shapes need to be modeled, and each sunflower instance is just

represented by its model ID and transformation matrix. This saves a lot of memory

compared to having to explicitly copy the sunflower geometry millions of times.

It is very simple to ray-trace instanced geometry: simply transform the ray

using the inverse of the instance transformation matrix, and test for intersection

with the original, untransformed object. If there is a hit, transform the hit point and

normal back using the instance transformation.

3.3.2 Ray reordering and shading caching

The Toro renderer [57] reordered the rays to increase the geometric coherency.

This made it possible to ray-trace scenes that are larger than the main memory of

the computer. Reordering the rays requires that the image contribution of each ray

is linear; this is true for real physical reflections but not generally true for the very

artistic programmable shaders used in movie production.

The Razor project by Stoll et al. [71] is inspired by the REYES algorithm used

for scanline rendering [18]. It shades entire grids of surface points at a time and

stores the view-independent parts of the shading results. If some of the following

rays hit the same surface patch, the shading results can be reused.

3.3.3 Geometric stand-ins

Wald et al. [75] demonstrated interactive rendering of complex scenes (for example

a Boeing 777 airplane with 350 million triangles) using approximate stand-ins for

42



geometry that has not been loaded yet. The precomputation of the stand-ins is

a rather lengthy process, but once it is completed, the scene can be ray-tracedinteractively.

3.3.4 Multiresolution tessellation

In practical applications, we have found it advantageous to tessellate curves, Bezier

patches, NURBS surfaces, subdivision surfaces, and any surface with displacement

instead of computing ray-surface intersections with numeric methods.

Patchification and tessellation

The surfaces are split into smaller surface patches of a manageable size — cor-

responding to tiling of textures. The tessellation rate for a directly visible surface

patch should depend on viewing distance and surface curvature, and optionally also

viewing angle. For reflections or shadows we can often use coarser tessellations.

Figure 3.3 shows an example of five tessellations of a surface patch; in this exam-

ple the finest tessellation rate is 14×11. The coarser levels consist of subsets of

the vertices of the finest tessellation. The coarsest tessellation is simply the four

corners of the patch. One can think of the various levels of tessellation as a MIP

map of tessellated geometry.

Figure 3.3: Multiresolution tessellation example for a surface patch: 14×11 quads,

7×6 quads, 4×3 quads, 2×2 quads, and 1 quad.

Multiresolution tessellation cache

Pharr and Hanrahan [55] cached tessellated geometry for displaced surfaces but did

not exploit multiresolution tessellation. We tessellate surface patches on demand

at the required resolution (then displace the vertices if appropriate) and store the

tessellations in a cache. Since the size of the tessellations differ so much, the cache

can store many more coarse tessellations than fine tessellations.

43



For ray intersection tests, we choose the tessellation where the quadrilaterals

are approximately the same size as the ray beam cross-section. We have observedthat accesses to the fine and medium tessellations are usually very coherent. The

accesses to the coarse tessellations are rather incoherent, but the cache capacity for

coarse tessellations is large and those tessellations are fast to recompute anyway.

The fine tessellations are only needed for directly visible geometry, for specular

reflections and refractions from flat surfaces, and for diffuse and ambient occlusion

rays near the ray origins. For all other rays, the ray beams are wide and the medium

and coarse tessellations are used.

Implementation details for our multiresultion tessellation cache can be found

in Christensen et al. [17, 16].

3.4 Parallel execution

Ray tracing seems very suitable for parallel speedups: the computations for each

pixel is independent of all other pixels. This has lead to the common belief that ray

tracing is “embarassingly parallel”. However, this is only true if the scene data fit

in main memory! If the scene is larger, great care must be taken to maintain and

exploit data access coherency. It pays off to arrange the execution order such that

subsequent rays tend to traverse the same geometry and access the same textures,

thus ensuring good cache behavior.

3.4.1 SIMD instructions

Modern CPUs have SIMD instructions (SSE on Intel, AltiVec on IBM/Motorola,

3dNow on AMD) that perform four operations in parallel. Wald et al. [79] utilized

these instructions to intersection-test four rays in parallel against one triangle. This

provides good speedups if the rays are coherent, and for visibility rays they reported

typical speedups around 3.5.

Another way to utilize the SIMD instructions is to intersection-test one ray

against four triangles in parallel. This gives good speedups if the triangles are

coherent — as they are if they come from adjacent positions on a tessellated surface

— and does not require the rays to be coherent. This was used by Christensen et

al. [16]. Another use of the SIMD instructions is to intersection-test all three slabs

of an axis-aligned bounding box in parallel.

44



3.4.2 Multiprocessors

Muuss [49] implemented parallel ray tracers both on multiprocessor machines with

shared memory and on distributed computers on a network.

Parker et al. [52] implemented a parallel ray tracer on an SGI Origin 2000

supercomputer with 64 processors. For maximum efficiency, the scene had to fit

within the local cache of each processor (4MB on the Origin), the shaders had

to be very simple, and the illumination could consist of only one light source.

For such relatively simple scenes they obtained interactive speeds, and despite the

restrictions the interactivity was quite an accomplishment.

Multiprocessors and multi-core architectures seem highly relevant for the fu-

ture of ray tracing.

3.4.3 Clusters of PCs

Wald et al. [78] and Kato [38] implemented ray tracers on clusters of standard

PCs. Wald’s renderer was real-time, while Kato’s Kilauea was a (non-interactive)

film-quality renderer. Both renderers used a brute-force approach: copy the scene

to a dozen or more PCs and send ray packets to each PC for intersection testing.

Kilauea could handle scenes larger than the memory of a single PC by dividing

the scene geometry up into as many parts as it took to store it. Each PC computed

ray-packet intersections for the parts of the scene that it contained.

3.5 Ray tracing in Pixar movies

As an example of the practical use of ray tracing in complex scenes, this section

describes how ray tracing has been used in the production of recent Pixar movies.

Pixar’s RenderMan renderer (PRMan) [74, 2] is based on the REYES scanline

rendering algorithm [18]. The REYES algorithm renders a small image tile at a

time; while rendering a tile it can ignore most of the data outside that tile — hence

it can deal with very complex scenes. Traditionally, shadows have been computed

with shadow maps [59] and reflections have been approximated with reflection

maps [7, 28].

Although still based on the REYES algorithm, we have extended PRMan with

with on-demand ray tracing [17, 16]. With PRMan’s hybrid rendering algorithm

there are no visibility rays, but ray tracing can be used to compute e.g. reflections,

45



shadows, and ambient occlusion. Thanks to the use of ray differentials and mul-

tiresolution texture and tessellation caches, very complex scenes can be ray-tracedwithout extraneous restrictions on the shaders, displacements, etc.

The first use of ray tracing in a Pixar movie (as far as I know) was for reflections

and refractions in a glass bottle in the movie A Bug’s Life. The ray tracing was

done with an external plug-in [2]. Since no algorithms were in place to deal with

ray tracing of very complex scenes, only a subset of the scene geometry was ray

traced.

The first wide-spread use of ray tracing at Pixar was for ambient occlusion in

the movie The Incredibles. It is interesting to note that the shaders at Pixar are so

complex and time-consuming that the time spent tracing rays is less than the time

spent evaluating shaders at the ray hit points. This is perhaps the main reason forthe popularity of ambient occlusion, both at Pixar and elsewhere: it is much faster

to compute ambient occlusion than e.g. ray-traced reflections. This despite the fact

that ambient occlusion usually requires tracing many more rays than reflections.

Ray tracing was also used in the movie Cars. In addition to ambient occlusion,

ray tracing was used to compute realistic reflections and to compute shadows in

large outdoor scenes with tiny shadow details. Figure 3.4 shows “beauty shots” of

two of the characters from Cars. Note in particular the reflection of the eyes in the

hoods.

(a) (b)

Figure 3.4: Cars with ray-traced reflections, shadows, and ambient occlusion:

(a) Luigi, a Fiat 500 “Topolino”; (b) Doc Hudson, a Hudson Hornet. (Copyright

c

2006 Disney/Pixar.)

For efficiency, the reflections in the Cars movie were usually limited to a single

level of reflection. There were only a few shots with two levels of reflection; they

are close-ups of chrome parts that needed to reflect themselves multiple times to

46



get the right look. Figure 3.5 shows an example, the rear chrome bumper on Doc

Hudson.

Figure 3.5: Doc Hudson’s chrome bumper with two levels of ray-traced reflection.

(Copyright c 2006 Disney/Pixar.)

Figure 3.6 shows all the main characters in Cars. (This is a section of a poster

that was originally rendered 3400 pixels wide.) This is a very complex scene with

many shiny cars. The shiny cars reflect other cars, as shown in the close-ups. The

image also shows ray-traced shadows and ambient occlusion.

Ray tracing is also being used in Pixar’s latest movie, Ratatouille — mostly for

ambient occlusion, reflections in pots and pans, and reflections and refractions in

glasses. Figure 3.7 shows an example of ray-traced reflections and refractions in

wine glasses.

47



Figure 3.6: The cast of Cars with two close-ups showing ray-traced reflections.


Figure 3.7: Ray-traced wine glasses from Ratatouille. (Copyright c 2007 Dis-

ney/Pixar.)

48



Chapter 4

A Practical Guide to Global

Illumination using Photon

Mapping

4.1 Photon tracing

The purpose of the photon tracing pass is to compute indirect illumination on dif-

fuse surfaces. This is done by emitting photons from the light sources, tracing them

through the scene, and storing them at diffuse surfaces.

4.1.1 Photon emission

This section describes how photons are emitted from a single light source and from

multiple light sources, and describes the use of projection maps which can increase

the emission efficiency considerably.

Emission from a single light source

The photons emitted from a light source should have a distribution corresponding

to the distribution of emissive power of the light source. This ensures that the

emitted photons carry the same flux — ie. we do not waste computational resources

on photons with low power.

Photons from a diffuse point light source are emitted in uniformly distributed

random directions from the point. Photons from a directional light are all emitted

in the same direction, but from origins outside the scene. Photons from a diffuse

49



square light source are emitted from random positions on the square, with direc-

tions limited to a hemisphere. The emission directions are chosen from a cosinedistribution: there is zero probability of a photon being emitted in the direction

parallel to the plane of the square, and highest probability of emission is in the

direction perpendicular to the square.

In general, the light source can have any shape and emission characteristics —

the intensity of the emitted light varies with both origin and direction. For example,

a (matte) light bulb has a nontrivial shape and the intensity of the light emitted

from it varies with both position and direction. The photon emission should follow

this variation, so in general, the probability of emission varies depending on the

position on the surface of the light source and the direction.

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Figure 4.1: Emission from light sources: (a) point light, (b) directional light,

(c) square light, (d) general light.

Figure 4.1 shows the emission from these different types of light sources.

The power (“wattage”) of the light source has to be distributed among the pho-

tons emitted from it. If the power of the light is P light and the number of emitted

photons is ne, the power of each emitted photon is

P photon =P lightne

. (4.1)

Pseudocode for a simple example of photon emission from a diffuse point light

source is given in Figure 4.2.

To further reduce variation in the computed indirect illumination (during ren-

dering), it is desirable that the photons are emitted as evenly as possible. This

can for example be done with stratification [?] or by using low-discrepancy quasi-

random sampling [?].

50



emit photons from diffuse point light()

ne = 0 number of emitted photons

while (not enough photons) do use simple rejection sampling to find diffuse photon direction

x = random number between -1 and 1

y = random number between -1 and 1

z = random number between -1 and 1

while ( x2+ y2 + z2 > 1 )

d = < x, y, z > p = light source position

trace photon from p in direction dne = ne + 1

scale power of stored photons with 1/ne

Figure 4.2: Pseudocode for emission of photons from a diffuse point light

Multiple lights

If the scene contains multiple light sources, photons should be emitted from each

light source. More photons should be emitted from brighter lights than from dim

lights, to make the power of all emitted photons approximately even. (The infor-

mation in the photon map is best utilized if the power of the stored photons is

approximately even). One might worry that scenes with many light sources would

require many more photons to be emitted than scenes with a single light source.

Fortunately, it is not so. In a scene with many light sources, each light contributes

less to the overall illumination, and typically fewer photons can be emitted from

each light. If, however, only a few light sources are important one might use an

importance sampling map [54] to concentrate the photons in the areas that are of

interest to the observer. The tricky part about using an importance map is that we

do not want to generate photons with energy levels that are too different since this

will require a larger number of photons in the radiance estimate (see section 4.3)

to ensure good quality.

51



Projection maps

In scenes with sparse geometry, many emitted photons will not hit any objects.

Emitting these photons is a waste of time. To optimize the emission, projection

maps can be used [?, ?]. A projection map is simply a map of the geometry as seen

from the light source. This map consists of many little cells. A cell is “on” if there

is geometry in that direction, and “off” if not. For example, a projection map is a

spherical projection of the scene for a point light, and it is a planar projection of the

scene for a directional light. To simplify the projection it is convenient to project

the bounding sphere around each object or around a cluster of objects [?]. This

also significantly speeds up the computation of the projection map since we do not

have to examine every geometric element in the scene. The most important aspect

about the projection map is that it gives a conservative estimate of the directions in

which it is necessary to emit photons from the light source. Had the estimate not

been conservative (e.g. we could have sampled the scene with a few photons first),

we could risk missing important effects, such as caustics.

The emission of photons using a projection map is very simple. One can either

loop over the cells that contain objects and emit a random photon into the direc-

tions represented by the cell. This method can, however, lead to slightly biased

results since the photon map can be “full” before all the cells have been visited.

An alternative approach is to generate random directions and check if the cell cor-

responding to that direction has any objects (if not a new random direction should

be tried). This approach usually works well, but it can be costly in sparse scenes.

For sparse scenes it is better to generate photons randomly for the cells which have

objects. A simple approach is to pick a random cell with objects and then pick a

random direction for the emitted photon for that cell [ ?]. In all circumstances it is

necessary to scale the energy of the stored photons based on the number of active

cells in the projection map and the number of photons emitted [?]. This leads to a

slight modification of formula 4.1:

P photon =P lightne

cells with objects

total number of cells. (4.2)

Another important optimization for the projection map is to identify objects

with specular properties (i.e. objects that can generate caustics) [?]. As it will be

described later, caustics are generated separately, and since specular objects often

are distributed sparsely it is very beneficial to use the projection map for caustics.

52



a

b

c

Figure 4.3: Photon paths in a scene (a “Cornell box” with a chrome sphere on left

and a glass sphere on right): (a) two diffuse reflections followed by absorption, (b) a

specular reflection followed by two diffuse reflections, (c) two specular transmissions

followed by absorption.

4.1.2 Photon tracing

Once a photon has been emitted, it is traced through the scene using photon tracing

(also known as “light ray tracing”, “backward ray tracing”, “forward ray tracing”,

and “backward path tracing”). Photon tracing works in exactly the same way as

ray tracing except for the fact that photons propagate flux whereas rays gather

radiance. This is an important distinction since the interaction of a photon with

a material can be different than the interaction of a ray. A notable example is

refraction where radiance is changed based on the relative index of refraction[ ?]

— this does not happen to photons.

When a photon hits an object, it can either be reflected, transmitted, or ab-

sorbed. Whether it is reflected, transmitted, or absorbed is decided probabilistically

based on the material parameters of the surface. The technique used to decide the

type of interaction is known as Russian roulette [?] — basically we roll a dice

and decide whether the photon should survive and be allowed to perform another

photon tracing step.

Examples of photon paths are shown in Figure 4.3.

53



Reflection, transmission, or absorption?

For a simple example, we first consider a monochromatic simulation. For a re-

flective surface with a diffuse reflection coefficient d and specular reflection coeffi-

cient s (with d + s ≤ 1) we use a uniformly distributed random variable ξ ∈ [0, 1]

(computed with for example drand48()) and make the following decision:

ξ ∈ [0, d] −→ diffuse reflection

ξ ∈]d, s + d] −→ specular reflection

ξ ∈]s + d, 1] −→ absorption

In this simple example, the use of Russian roulette means that we do not have to

modify the power of the reflected photon — the correctness is ensured by averaging

several photon interactions over time. Consider for example a surface that reflects

50% of the incoming light. With Russian roulette only half of the incoming photons

will be reflected, but with full energy. For example, if you shoot 1000 photons at

the surface, you can either reflect 1000 photons with half the energy or 500 photons

with full energy. It can be seen that Russian roulette is a powerful technique for

reducing the computational requirements for photon tracing.

With more color bands (for example RGB colors), the decision gets slightly

more complicated. Consider again a surface with some diffuse reflection and some

specular reflection, but this time with different reflection coefficients in the three

color bands. The probabilities for specular and diffuse reflection can be based on

the total energy reflected by each type of reflection or on the maximum energy

reflected in any color band. If we base the decision on maximum energy, we can

for example compute the probability P d for diffuse reflection as

P d =max(drP r, dgP g, dbP b)

max(P r, P g, P b)

where (dr, dg, db) are the diffuse reflection coefficients in the red, green, and blue

color bands, and (P r, P g, P b) are the powers of the incident photon in the same

three color bands.

Similarly, the probability P s for specular reflection is

P s =max(srP r, sgP g, sbP b)

max(P r, P g, P b)

where (sr, sg, sb) are the specular reflection coefficients.

54



The probability of absorbtion is P a = 1 − P d − P s. With these probabilities,

the decision of which type of reflection or absorption should be chosen takes thefollowing form:

ξ ∈ [0, P d] −→ diffuse reflection

ξ ∈]P d, P s + P d] −→ specular reflection

ξ ∈]P s + P d, 1] −→ absorption

The power of the reflected photon needs to be adjusted to account for the proba-

bility of survival. If, for example, specular reflection was chosen in the example

above, the power P refl of the reflected photon is:

P refl,r = P inc,r sr/P sP refl,g = P inc,g sg/P sP refl,b = P inc,b sb/P s

where P inc is the power of the incident photon.

The computed probabilities again ensure us that we do not waste time emitting

photons with very low power.

It is simple to extend the selection scheme to also handle transmission, to han-

dle more than three color bands, and to handle other reflection types (for example

glossy and directional diffuse).

Why Russian roulette?

Why do we go through this effort to decide what to do with a photon? Why not justtrace new photons in the diffuse and specular directions and scale the photon energy

accordingly? There are two main reasons why the use of Russian roulette is a very

good idea. Firstly, we prefer photons with similar power in the photon map. This

makes the radiance estimate much better using only a few photons. Secondly, if

we generate, say, two photons per surface interaction then we will have 28 photons

after 8 interactions. This means 256 photons after 8 interactions compared to 1

photon coming directly from the light source! Clearly this is not good. We want at

least as many photons that have only 1–2 bounces as photons that have made 5–8

bounces. The use of Russian roulette is therefore very important in photon tracing.

There is one caveat with Russian roulette. It increases variance on the solution.

Instead of using the exact values for reflection and transmission to scale the photon

energy we now rely on a sampling of these values that will converge to the correct

result as enough photons are used.

Details on photon tracing and Russian roulette can be found in [?, ?, 26].

55



(a) (b)

Figure 4.4: “Cornell box” with glass and chrome spheres: (a) ray traced image (di-

rect illumination and specular reflection and transmission), (b) the photons in the

corresponding photon map.

4.1.3 Photon storing

This section describes which photon-surface interactions are stored in the photon

map. It also describes in more detail the photon map data structure.

Which photon-surface interactions are stored?

Photons are only stored where they hit diffuse surfaces (or, more precisely, non-

specular surfaces). The reason is that storing photons on specular surfaces does

not give any useful information: the probability of having a matching incoming

photon from the specular direction is zero, so if we want to render accurate specular

reflections the best way is to trace a ray in the mirror direction using standard ray

tracing. For all other photon-surface interactions, data is stored in a global data

structure, the photon map. Note that each emitted photon can be stored several

times along its path. Also, information about a photon is stored at the surface

where it is absorbed if that surface is diffuse.

For each photon-surface interaction, the position, incoming photon power, and

incident direction are stored. (For practical reasons, there is also space reserved for

a flag with each set of photon data. The flag is used during sorting and look-up in

56



the photon map. More on this in the following.)

As an example, consider again the simple scene from Figure 4.3, a “Cornellbox” with two spheres. Figure 4.4(a) shows a traditional ray traced image (direct

illumination and specular reflection and transmission) of this scene. Figure 4.4(b)

shows the photons in the photon map generated for this scene. The high concen-

tration of photons under the glass sphere is caused by focusing of the photons by

the glass sphere.

Data structure

Expressed in C the following structure is used for each photon [?]:

struct photon float x,y,z; // position

char p[4]; // power packed as 4 chars

char phi, theta; // compressed incident direction

short flag; // flag used in kdtree

The power of the photon is represented compactly as 4 bytes using Ward’s

packed rgb-format [80]. If memory is not of concern one can use 3 floats to store

the power in the red, green, and blue color band (or, in general, one float per color

band if a spectral simulation is performed).

The incident direction is a mapping of the spherical coordinates of the photon

direction to 65536 possible directions. They are computed as:

phi = 255 * (atan2(dy,dx)+PI) / (2*PI)

theta = 255 * acos(dx) / PI

where atan2 is from the standard C library. The direction is used to compute

the contribution for non-Lambertian surfaces [?], and for Lambertian surfaces it

is used to check if a photon arrived at the front of the surface. Since the photon

direction is used often during rendering it pays to have a lookup table that maps the

theta, phi direction to three floats directly instead of using the formula for spherical

coordinates which involves the use of the costly cos() and sin() functions.

A minor note is that the flag in the structure is a short. Only 2 bits of this flag

are used (this is for the splitting plane axis in the kd-tree), and it would be possible

to use just one byte for the flag. However for alignment reasons it is preferable to

57



have a 20 byte photon rather than a 19 byte photon — on some architectures it is

even a necessity since the float-value in subsequent photons must be aligned on a4 byte address.

We might be able to compress the information more by using the fact that we

know the cube in which the photon is located. The position is, however, used very

often when the photons are processed and by using standard float we avoid the

overhead involved in extracting the true position from a specialized format.

During the photon tracing pass the photon map is arranged as a flat array of

photons. For efficiency reasons this array is re-organized into a balanced kdtree

before rendering as explained in section 4.2.

4.1.4 Extension to participating media

Up to this point, all photon interactions have been assumed to happen at object

surfaces; all volumes were implicitly assumed to not affect the photons. However,

it is simple to extend the photon map method to handle participating media, i.e.

volumes that participate in the light transport. In scenes with participating media,

the photons are stored within the media in a seperate volume photon map [?].

Photon emission, tracing, and storage

Photons can be emitted from volumes as well as from surfaces and points. For

example, the light from a candle flame can be simulated by emitting photons froma flame-shaped volume.

When a photon travels through a participating medium, it has a certain proba-

bility of being scattered or absorped in the medium. The probability depends on the

density of the medium and on the distance the photon travels through the medium:

the denser the medium, the shorter the average distance before a photon interaction

happens. Photons are stored at the positions where a scattering event happens. The

exception is photons that come directly from the light source since direct illumi-

nation is evaluated using ray tracing. This separation was introduced in [?] and it

allows us to compute the in-scattered radiance in a medium simply by a lookup in

the photon map.

As an example, consider a glass sphere in fog illuminated by directional light.

Figure 4.5(a) shows a schematic diagram of the photon paths as photons are being

focused by refraction in the glass sphere. Figure 4.5(b) shows the caustic photons

stored in the photon map.

58



Figure 4.5: Sphere in fog: (a) schematic diagram of light paths, (b) the caustic pho-

tons in the photon map.

Multiple scattering, anisotropic scattering, and non-homogeneous media

The simple example above only shows the photon interaction in the fog after refrac-

tion by the glass sphere, and the photon paths are terminated at the first scattering

event. General multiple scattering is simulated simply by letting the photons scatter

everywhere and continuously after the first interaction. The path can be terminated

using Russian roulette.

The fog in the example has uniform density, but it is not difficult to handle

media with nonuniform density (aka. nonhomogeneous media), since we use ray

marching to integrate the properties of the medium. A simple ray marcher works

by dividing the medium into little steps [?]. The accumulated density (integrated

extinction coefficient) is updated at each step, and based on a precomputed proba-

bility it is determined whether the photon should be absorbed, scattered, or whetheranother step is necessary.

For more complicated examples of scattering in participating media, including

anisotropic and nonhomogeneous media and complex geometry, see [?].

59



4.1.5 Three photon maps

For efficiency reasons, it pays off to divide the stored photons into three photon

maps:

Caustic photon map: contains photons that have been through at least one spec-

ular reflection before hitting a diffuse surface: LS +D.

Global photon map: an approximate representation of the global illumination so-

lution for the scene for all diffuse surfaces: LS |D|V ∗D

Volume photon map: indirect illumination of a participating medium:

LS |D|V +V .

Here, we used the grammar from [?] to describe the photon paths: L means emis-

sion from the light source, S is specular reflection or transmission, D is diffuse (ie.

non-specular) reflection or transmission, and V is volume scattering. The notation

x|y|z means “one of x, y, or z”, x+ means one or several repeats of x, and x∗

means zero or several repeats of x.

The reason for keeping three separate photon maps will become clear in sec-

tion 4.4. A separate photon tracing pass is performed for the caustic photon map

since it should be of high quality and therefore often needs more photons than the

global photon map and the volume photon map.

The construction of the photon maps is most easily achieved by using two sep-

arate photon tracing steps in order to build the caustics photon map and the global

photon map (including the volume photon map). This is illustrated in Figure 4.6

for a simple test scene with a glass sphere and 2 diffuse walls. Figure 4.6(a) shows

the construction of the caustics photon map with a dense distribution of photons,

and Figure 4.6(b) shows the construction of the global photon map with a more

coarse distribution of photons.

4.2 Preparing the photon map for rendering

Photons are only generated during the photon tracing pass — in the rendering pass

the photon map is a static data structure that is used to compute estimates of the

incoming flux and the reflected radiance at many points in the scene. To do this it

is necessary to locate the nearest photons in the photon map. This is an operation

60



(a) (b)

Figure 4.6: Building (a) the caustics photon map and (b) the global photon map.

that is done extremely often, and it is therefore a good idea to optimize the repre-

sentation of the photon map before the rendering pass such that finding the nearest

photons is as fast as possible.

First, we need to select a good data structure for representing the photon map.

The data structure should be compact and at the same time allow for fast nearest

neighbor searching. It should also be able to handle highly non-uniform distribu-

tions — this is very often the case in the caustics photon map. A natural candidate

that handles these requirements is a balanced kd-tree [?]. Examples of using a

balanced versus an unbalanced kd-tree can be found in [?].

4.2.1 The balanced kd-tree

The time it takes to locate one photon in a balanced kd-tree has a worst time per-

formance of O(log N ), where N is the number of photons in the tree. Since the

photon map is created by tracing photons randomly through a model one might

think that a dynamically built kd-tree would be quite well balanced already. How-

ever, the fact that the generation of the photons at the light source is based on the

projection map combined with the fact that models often contain highly directional

reflectance models easily results in a skewed tree. Since the tree is created only

once and used many times during rendering it is quite natural to consider balanc-

ing the tree. Another argument that is perhaps even more important is the fact that

a balanced kd-tree can be represented using a heap-like data-structure [?] which

means that explicitly storing the pointers to the sub-trees at each node is no longer

necessary. (Array element 1 is the tree root, and element i has element 2i as left

child and element 2i + 1 as right child.) This can lead to considerable savings in

61



kdtree *balance( points )

Find the cube surrounding the pointsSelect dimension dim in which the cube is largest

Find median of the points in dims1 = all points below median

s2 = all points above median

node = median

node.left = balance( s1 )

node.right = balance( s2 )

return node

Figure 4.7: Pseudocode for balancing the photon map

memory when a large number of photons is used.

4.2.2 Balancing

Balancing a kd-tree is similar to balancing a binary tree. The main difference is the

choice at each node of a splitting dimension. When a splitting dimension of a set is

selected, the median of the points in that dimension is chosen as the root node of the

tree representing the set and the left and right subtrees are constructed from the two

sets separated by the median point. The choice of a splitting dimension is based

on the distribution of points within the set. One might use either the variance or

the maximum distance between the points as a criterion. We prefer a choice based

upon maximum distance since it can be computed very efficiently (even though

a choice based upon variance might be slightly better). The splitting dimension is

thus chosen as the one which has the largest maximum distance between the points.

Figure 4.7 contains a pseudocode outline for the balancing algorithm [?].

To speed up the balancing process it is convenient to use an array of pointers

to the photons. In this way only pointers needs to be shuffled during the median

search. An efficient median search algorithm can be found in most textbooks on

algorithms — see for example [?] or [?].

The complexity of the balancing algorithm is O(N log N ) where N is the num-

ber of photons in the photon map. In practice, this step only takes a few seconds

even for several million photons.

62



4.3 The radiance estimate

A fundamental component of the photon map method is the ability to compute

radiance estimates at any non-specular surface point in any given direction.

4.3.1 Radiance estimate at a surface

The photon map can be seen as a representation of the incoming flux; to com-

pute radiance we need to integrate this information. This can be done using the

expression for reflected radiance:

Lr(x, ω) =

Ωx

f r(x, ω, ω)Li(x, ω)|nx · ω| dωi , (4.3)

where Lr is the reflected radiance at x in direction ω. Ωx is the (hemi)sphere of

incoming directions, f r is the BRDF (bidirectional reflectance distribution func-

tion) [?] at x and Li is the incoming radiance. To evaluate this integral we need

information about the incoming radiance. Since the photon map provides informa-

tion about the incoming flux we need to rewrite this term. This can be done using

the relationship between radiance and flux:

Li(x, ω) =

d2Φi(x, ω)

cos θi dωi dAi, (4.4)

and we can rewrite the integral as

Lr(x, ω) =

Ωx

f r(x, ω, ω)d2Φi(x, ω

)

cos θi dωi dAi

|nx · ω| dωi

=

Ωx

f r(x, ω, ω)d2Φi(x, ω

)

dAi. (4.5)

The incoming flux Φi is approximated using the photon map by locating the n

photons that has the shortest distance to x. Each photon p has the power ∆Φ p( ω p)

and by assuming that the photons intersects the surface at x we obtain

Lr(x, ω) ≈n

p=1

f r(x, ω p, ω) ∆Φ p(x, ω p)∆A

. (4.6)

The procedure can be imagined as expanding a sphere aroundx until it contains

n photons (see Figure 4.8) and then using these n photons to estimate the radiance.

63



L L

Figure 4.9: Using a sphere (left) and using a disc (right) to locate the photons.

result where N is the number of photons in the photon map:

limN →∞

1

πr2

N α p=1

f r(x, ω p, ω)∆Φ p(x, ω p) = Lr(x, ω) for α ∈]0, 1[ . (4.9)

This formulation applies to all points x located on a locally flat part of a surface for

which the BRDF, does not contain the Dirac delta function (this excludes perfect

specular reflection). The principle in equation 4.9 is that not only will an infinite

amount of photons be used to represent the flux within the model but an infinite

amount of photons will also be used to estimate radiance and the photons in the

estimate will be located within an infinitesimal sphere. The different degrees of in-

finity are controlled by the term N α

where α ∈]0, 1[. This ensures that the numberof photons in the estimate will be infinitely fewer than the number of photons in

the photon map.

Equation 4.9 means that we can obtain arbitrarily good radiance estimates by

just using enough photons! In finite element based approaches it is more compli-

cated to obtain arbitrary accuracy since the error depends on the resolution of the

mesh, the resolution of the directional representation of radiance and the accuracy

of the light simulation.

Figure 4.8 shows how locating the nearest photons is similar to expanding a

sphere around x and using the photons within this sphere. It is possible to use

other volumes than the sphere in this process. One might use a cube instead, a

cylinder or perhaps a disc. This could be useful to either obtain an algorithm that

is faster at locating the nearest photons or perhaps more accurate in the selection of

photons. If a different volume is used then ∆A in equation 4.6 should be replaced

by the area of the intersection between the volume and the tangent plane touching

65



the surface at x. The sphere has the obvious advantage that the projected area and

the distance computations are very simple and thus efficiently computed. A moreaccurate volume can be obtained by modifying the sphere into a disc (ellipsoid)

by compressing the sphere in the direction of the surface normal at x (shown in

Figure 4.9) [?]. The advantage of using a disc would be that fewer “false photons”

are used in the estimate at edges and in corners. This modification works pretty

well at the edges in a room, for instance, since it prevents photons on the walls to

leak down to the floor. One issue that still occurs, however, is that the area estimate

might be wrong or photons may leak into areas where they do not belong. This

problem is handled primarily by the use of filtering.

4.3.2 Filtering

If the number of photons in the photon map is too low, the radiance estimates be-

comes blurry at the edges. This artifact can be pleasing when the photon map is

used to estimate indirect illumination for a distribution ray tracer (see section 4.4

and Figure 4.15) but it is unwanted in situations where the radiance estimate rep-

resents caustics. Caustics often have sharp edges and it would be nice to preserve

these edges without requiring too many photons.

To reduce the amount of blur at edges, the radiance estimate is filtered. The

idea behind filtering is to increase the weight of photons that are close to the point

of interest, x. Since we use a sphere to locate the photons it would be natural to

assume that the filters should be three-dimensional. However, photons are stored

at surfaces which are two-dimensional. The area estimate is also based on the

assumption that photons are located on a surface. We therefore need a 2d-filter

(similar to image filters) which is normalized over the region defined by the pho-

tons.

The idea of filtering caustics is not new. Collins [?] has examined several

filters in combination with illumination maps. The filters we have examined are

two radially symmetric filters: the cone filter and the Gaussian filter [?], and the

specialized differential filter introduced in [?]. For examples of more advanced

filters see Myszkowski et al. [?].

66



The cone filter

The cone-filter [?] assigns a weight, w pc, to each photon based on the distance, d p,

between x and the photon p. This weight is:

w pc = 1− d pk r

, (4.10)

where k ≥ 1 is a filter constant characterizing the filter and r is the maximum

distance. The normalization of the filter based on a 2d-distribution of the photons

is 1− 2

3kand the filtered radiance estimate becomes:

Lr(x, ω) ≈

N

p=1

f r(x, ω p, ω)∆Φ p(x, ω p)w pc

(1− 2

3k)πr2

. (4.11)

The Gaussian filter

The Gaussian filter [?] has previously been reported to give good results when

filtering caustics in illumination maps [?]. It is easy to use the Gaussian filter with

the photon map since we do not need to warp the filter to some surface function.

Instead we use the assumption about the locally flat surfaces and we can use a

simple image based Gaussian filter [?] and the weight w pg of each photon becomes

w pg = α

1− 1− e−β

d2p

2r2

1 − e−β

, (4.12)

where d p is the distance between the photon p and x and α = 0.918 and β = 1.953

(see [?] for details). This filter is normalized and the only change to equation 4.8

is that each photon contribution is multiplied by w pg:

Lr(x, ω) ≈N

p=1

f r(x, ω p, ω)∆Φ p(x, ω p)w pg . (4.13)

Differential checking

In [?] it was suggested to use a filter based on differential checking. The idea is to

detect regions near edges in the estimation process and use less photons in these

67



regions. In this way we might get some noise in the estimate but that is often

preferable to blurry edges.The radiance estimate is modified based on the following observation: when

adding photons to the estimate, near an edge the changes of the estimate will be

monotonic. That is, if we are just outside a caustic and we begin to add photons

to the estimate (by increasing the size of the sphere centered at x that contains the

photons), then it can be observed that the value of the estimate is increasing as we

add more photons; and vice versa when we are inside the caustic. Based on this

observation, differential checking can be added to the estimate — we stop adding

photons and use the estimate available if we observe that the estimate is either

constantly increasing or decreasing as more photons are added.

4.3.3 The radiance estimate in a participating medium

For the radiance estimate presented so far we have assumed that the photons are

located on a surface. For photons in a participating medium the formula changes

to [?]:

Li(x, ω) =

Ω

f (x, ω, ω)L(x, ω) dω

=

Ω

f (x, ω, ω)d2Φ(x, ω)

σs(x) dω dV dω

=

1

σs(x) Ωf (x, ω

, ω)

d2Φ(x, ω)

dV

≈ 1

σs(x)

n p=1

f (x, ω p, ω)∆Φ p(x, ω p)

43πr3

, (4.14)

where Li is the in-scattered radiance, and the volume dV = 43πr3 is the volume of

the sphere containing the photons. σs(x) is the scattering coefficient at x and f is

the phase-function.

4.3.4 Locating the nearest photons

Efficiently locating the nearest photons is critical for good performance of the pho-

ton map algorithm. In scenes with caustics, multiple diffuse reflections, and/or

participating media there can be a large number of photon map queries.

Fortunately the simplicity of the kd-tree permits us to implement a simple but

quite efficient search algorithm. This search algorithm is a straight forward exten-

68



sion of standard search algorithms for binary trees [?, ?, ?]. It is also related to

range searching where kd-trees are commonly used as they have optimal storageand good performance [?]. The nearest neighbors query for kd-trees has been de-

scribed extensively in several publications by Bentley et al. including [?, ?, ?, ?].

More recent publications include [?, ?]. Some of these papers go beyond our de-

scription of a nearest neighbors query by adding modifications and extensions to

the kd-tree to further reduce the cost of searching. We do not implement these

extensions because we want to maintain the low storage overhead of the kd-tree as

this is an important aspect of the photon map.

Locating the nearest neighbors in a kd-tree is similar to range searching [ ?] in

the sense that we want to locate photons within a given volume. For the photon map

it makes sense to restrict the size of the initial search range since the contributionfrom a fixed number of photons becomes small for large regions. This simple

observation is particularly important for caustics since they often are concentrated

in a small region. A search algorithm that does not limit the search range will be

slow in such situations since a large part of the kd-tree will be visited for regions

with a sparse number of photons.

A generic nearest neighbors search algorithm begins at the root of the kd-tree,

and adds photons to a list if they are within a certain distance. For the n nearest

neighbors the list is sorted such that the photon that is furthest away can be deleted

if the list contains n photons and a new closer photon is found. Instead of naive

sorting of the full list it is better to use a max-heap [ ?, ?, ?]. A max-heap (also

known as a priority queue) is a very efficient way of keeping track of the element

that is furthest away from the point of interest. When the max-heap is full, we can

use the distance d to the root element (ie. the photon that is furthest away) to adjust

the range of the query. Thus we skip parts of the kd-tree that are further away than

d.

Another simple observation is that we can use squared distances — we do not

need the real distance. This removes the need of a square root calculation per

distance check.

The pseudo-code for the search algorithm is given in Figure 4.10. A simple

implementation of this routine is available with source code at [?].

For this search algorithm it is necessary to provide an initial maximum search

radius. A well-chosen radius allows for good pruning of the search reducing the

number of photons tested. A maximum radius that is too low will on the other hand

introduce noise in the photon map estimates. The radius can be chosen based on an

69



given the photon map, a position x and a max search distance d2

this recursive function returns a heap h with the nearest photons.Call with locate photons(1) to initiate search at the root of the kd-tree

locate photons( p ) if ( 2 p + 1 < number of photons )

examine child nodes

Compute distance to plane (just a subtract)

δ = signed distance to splitting plane of node nif (δ < 0)

We are left of the plane - search left subtree first

locate photons( 2 p )

if ( δ2 < d2 )

locate photons( 2 p + 1 ) check right subtree

else

We are right of the plane - search right subtree first

locate photons( 2 p + 1 )

if ( δ2 < d2 )

locate photons( 2 p ) check left subtree

Compute true squared distance to photon

δ2 = squared distance from photon p to xif ( δ2 < d2

) Check if the photon is close enough?

insert photon into max heap hAdjust maximum distance to prune the search

d2 = squared distance to photon in root node of h

Figure 4.10: Pseudocode for locating the nearest photons in the photon map

error metric or the size of the scene. The error metric could for example take the

average energy of the stored photons into account and compute a maximum radius

from that assuming some allowed error in the radiance estimate.

A few extra optimizations can be added to this routine, for example a delayed

construction of the max heap to the time when the number of photons needed has

been found. This is particularly useful when the requested number of photons is

large.

Nathan Kopp has implemented a slightly different optimization in an extended

version of the Persistence Of Vision Ray Tracer (POV) called MegaPov (avail-

70



Figure 4.11: Tracing a ray through a pixel.

able at [?]). In his implementation the initial maximum search radius is set to a

very low value. If this value turns out to be too low, another search is performed

with a higher maximum radius. He reports good timings and results from this

technique [?].

Another change to the search routine is to use the disc check as described ear-

lier. This is useful to avoid incorrect color bleeding and particularly helpful if the

gathering step is not used and the photons are visualized directly.

4.4 Rendering

Given the photon map and the ability to compute a radiance estimate from it, we

can proceed with the rendering pass. The photon map is view independent, and

therefore a single photon map constructured for an environment can be utilized to

render the scene from any desired view. There are several different ways in which

the photon map can be visualized. A very fast visualization technique has been

presented by Myszkowski et al. [?, ?] where photons are used to compute radiosity

values at the vertices of a mesh.

In this note we will focus on the full global illumination approach as presented

in [?]. Initially we will ignore the presence of participating media; at the end of the

note we have added some notes for this case.

The final image is rendered using distribution ray tracing in which the pixel ra-

diance is computed by averaging a number of sample estimates. Each sample con-

sists of tracing a ray from the eye through a pixel into the scene (see Figure 4.11).

71



The radiance returned by each ray equals the outgoing radiance in the direction of

the ray leaving the point of intersection at the first surface intersected by the ray.The outgoing radiance, Lo, is the sum of the emitted, Le, and the reflected radiance

Lo(x, ω) = Le(x, ω) + Lr(x, ω) , (4.15)

where the reflected radiance, Lr, is computed by integrating the contribution from

the incoming radiance, Li,

Lr(x, ω) =

Ωx

f r(x, ω, ω)Li(x, ω)cos θi dω

i , (4.16)

where f r is the bidirectional reflectance distribution function (BRDF), and Ωx is

the set of incoming directions around x.

Lr can be computed using Monte Carlo integration techniques like path tracing

and distribution ray tracing. These methods are very costly in terms of rendering

time and a more efficient approach can be obtained by using the photon map in

combination with our knowledge of the BRDF and the incoming radiance.

The BRDF is separated into a sum of two components: A specular/glossy, f r,s,

and a diffuse, f r,d

f r(x, ω, ω) = f r,s(x, ω, ω) + f r,d(x, ω, ω) . (4.17)

The incoming radiance is classified using three components:

• Li,l(x, ω) is direct illumination by light coming from the light sources.

• Li,c(x, ω) is caustics — indirect illumination from the light sources via

specular reflection or transmission.

• Li,d(x, ω) is indirect illumination from the light sources which has been

reflected diffusely at least once.

The incoming radiance is the sum of these three components:

Li(x, ω) = Li,l(x, ω

) + Li,c(x, ω) + Li,d(x, ω) . (4.18)

By using the classifications of the BRDF and the incoming radiance we can

72



split the expression for reflected radiance into a sum of four integrals:

Lr(x, ω) =

Ωx

f r(x, ω, ω)Li(x, ω)cos θi dω

i

=

Ωx

f r(x, ω, ω)Li,l(x, ω)cos θi dω

i +

Ωx

f r,s(x, ω, ω)(Li,c(x, ω) + Li,d(x, ω))cosθi dωi +

Ωx

f r,d(x, ω, ω)Li,c(x, ω)cos θi dωi +

Ωx

f r,d(x, ω, ω)Li,d(x, ω)cos θi dωi . (4.19)

This is the equation used whenever we need to compute the reflected radiance

from a surface. In the following sections we discuss the evaluation of each of

the integrals in the equation in more detail. We distinguish between two different

situations: an accurate and an approximate.

The accurate computation is used if the surface is seen directly by the eye or

perhaps via a few specular reflections. It is also used if the distance between the ray

origin and the intersection point is below a small threshold value — to eliminate

potential inaccurate color bleeding effects in corners. The approximate evaluation

is used if the ray intersecting the surface has been reflected diffusely since it left

the eye or if the ray contributes only little to the pixel radiance.

4.4.1 Direct illumination

Direct illumination is given by the term

Ωx

f r(x, ω, ω)Li,l(x, ω)cos θi dω

i ,

and it represents the contribution to the reflected radiance due to direct illumina-

tion. This term is often the most important part of the reflected radiance and it

has to be computed accurately since it determines light effects to which the eye is

highly sensitive such as shadow edges.

Computing the contribution from the light sources is quite simple in ray tracing

based methods. At the point of interest shadow rays are sent towards the light

sources to test for possible occlusion by objects. This is illustrated in Figure 4.12.

If a shadow ray does not hit an object the contribution from the light source is

73



Figure 4.12: Accurate evaluation of the direct illumination.

included in the integral otherwise it is neglected. For large area light sources several

shadow rays are used to properly integrate the contribution and correctly render

penumbra regions. This strategy can however be very costly since a large number

of shadow rays is needed to properly integrate the direct illumination.

Using a derivative of the photon map method we can compute shadows more

efficiently using shadow photons [?]. This approach can lead to considerable

speedups in scenes with large penumbra-regions that are normally very costly to

render using standard ray tracing. The approach is stochastic though, so it might

miss shadows from small objects in case these aren’t intersected by any photons.

This is a problem with all techniques that use stochastic evaluation of visibility.

The approximate evaluation is simply the radiance estimate obtained from the

global photon map (no shadow rays or light source evaluations are used). This is

seen in Figure 4.15 where the global photon map is used in the evaluation of the

incoming light for the secondary diffuse reflection.

4.4.2 Specular and glossy reflection

Specular and glossy reflection is computed by evaluation of the term

Ωx

f r,s(x, ω, ω)(Li,c(x, ω) + Li,d(x, ω))cosθi dωi .

The photon map is not used in the evaluation of this integral since it is strongly

74



Figure 4.13: Rendering specular and glossy reflections.

dominated by f r,s which has a narrow peak around the mirror direction. Using the

photon map to optimize the integral would require a huge number of photons in

order to make a useful classification of the different directions within the narrow

peak of f r,s. To save memory this strategy is not used and the integral is evaluated

using standard Monte Carlo ray tracing optimized with importance sampling based

on f r,s. This is still quite efficient for glossy surfaces and the integral can in most

situations be computed using only a small number of sample rays.

This is illustrated in Figure 4.13.

4.4.3 Caustics

Caustics are represented by the integral

Ωx

f r,d(x, ω, ω)Li,c(x, ω)cos θi dωi .

The evaluation of this term is dependent on whether an accurate or an approximate

computation is required. In the accurate computation, the term is solved by us-

ing a radiance estimate from the caustics photon map. The number of photons in

the caustics photon map is high and we can expect good quality of the estimate.

Caustics are never computed using Monte Carlo ray tracing since this is a very in-

efficient method when it comes to rendering caustics. The approximate evaluation

of the integral is included in the radiance estimate from the global photon map.

75



Figure 4.14: Rendering caustics.


4.4.4 Multiple diffuse reflections

The last term in equation 4.19 is Ωx

f r,d(x, ω, ω)Li,d(x, ω)cos θi dωi .

This term represents incoming light that has been reflected diffusely at least once

since it left the light source. The light is then reflected diffusely by the surface

(using f r,d). Consequently the resulting illumination is very “soft”.

The approximate evaluation of this integral is a part of the radiance estimate

based on the global photon map.

The accurate evaluation of the integral is calculated using Monte Carlo ray

tracing optimized using the BRDF with an estimate of the flux as described in [?].

An important optimization at Lambertian surfaces is the use of Ward’s irradiance

gradient caching scheme [82, ?]. This means that we only compute indirect illu-

mination on Lambertian surfaces if we cannot interpolate with sufficient accuracy

from previously computed values. The advantage of using the photon map com-

pared to just using the irradiance gradient caching method is that we avoid having

to trace multiple bounces of indirect illumination and we can use the information

in the photon map to concentrate our samples into the important directions.

76




Figure 4.15: Computing indirect diffuse illumination with importance sampling.

4.4.5 Participating media

In the presence of participating media we can still use the framework as presented

so far. The main difference is that we need to take the media into account as we

trace rays through the scene. This can be done quite efficiently using ray marchingand the volume radiance estimate as described in [?].

4.4.6 Why distribution ray tracing?

The rendering method presented here is a combination of many algorithms. In or-

der to render accurate images without using too many photons a distribution ray

tracer is used to compute illumination seen directly by the eye. One might con-

sider visualizing the global photon map directly, and this would indeed be a full

global illumination solution (it would be similar to the density estimation approach

presented in [?]). The problem with this approach is that an accurate solution re-

quires a large number of photons. Significantly fewer photons are necessary when

a distribution ray tracer is used to evaluate the first diffuse reflection. If a blurry

solution is not a problem (for example for previewing) then a direct visualization

of the photon map can be used. For more accurate results it is often necessary to

77



use more than 1000 photons in the radiance estimate (see the results section for

some examples).

78



4.5 Examples

In this section we present some examples of scenes rendered using photon maps.

Please see the photon map web-page at http://www.gk.dtu.dk/photonmap

for the latest results. Also refer to the papers included in these notes for more ex-

amples.

All the images have been rendered using the Dali rendering program. Dali

is an extremely flexible renderer that supports ray tracing with global illumination

and participating media. The global illumination simulation code based on photon

maps is a module in Dali that is loaded at runtime. All material and geometry

code is also represented via modules that are loaded at runtime. Dali is multi-

threaded and all images have been rendered on a dual PentiumII-400 PC running

Linux. The width of each image is 1024 pixels and 4 samples per pixel have been

used.

4.5.1 The Cornell box

Most global illumination papers feature a simulation of the Cornell box, and so

does this note. Since we are not limited to radiosity our version of the Cornell

box is slightly different. It has a mirror sphere and a glass sphere instead of the

two cubes featured in the original Cornell box (the original Cornell box can be

found at http://www.graphics.cornell.edu/online/box/ ). Clas-

sic radiosity methods have difficulties handling curved specular objects, but raytracing methods (including the photon map method) have no problems with these.

Ray tracing

The image in Figure 4.16 shows the ray traced version of the Cornell box. Notice

the sharp shadows and the black ceiling of the box due to lack of area lights and

global illumination. Rendering time was 3.5 seconds.

Ray tracing with soft shadows

In Figure 4.17 soft shadows have been added. It has been reported that some people

associate soft shadows with global illumination, but in the Cornell box example it

is still obvious that something is missing. The ceiling is still black. Rendering time

was 21 seconds.

79



Figure 4.16: Ray traced Cornell box with sharp shadows.

Adding caustics

The image in Figure 4.18 includes the caustics photon map. Notice the bright spotbelow the glass sphere and on the right wall (due to light reflected of the mirror

sphere and transmitted through the glass sphere). Also notice the faint illumination

of the ceiling. The caustics photon map has 50000 photons and the estimate uses

up to 60 photons. Photon tracing took 2 seconds. Rendering time was 34 seconds.

We did not use any filtering of the caustics photons. A maximum search distance

of 0.15 was used for the caustics photon map (the depth of the Cornell box is 5

units). Using a search distance of 0.5 increased the rendering time to 42 seconds.

For an unlimited initial search radius the rendering time was 43 seconds. The

computed images looked very similar. The faint illumination of the ceiling is a

caustic (created by the bright caustic on the floor) — it becomes a little softer with

the increased search radius. For a search radius of 0.01 the caustics became more

noisy, and the rendering time was 25 seconds. For other scenes where the caustics

are more localized the influence of the maximum search radius on the rendering

time can be more dramatic than for the Cornell box.

80



Figure 4.17: Ray traced Cornell box with soft shadows.

Figure 4.18: Cornell box with caustics.

81



Figure 4.19: Cornell box with global illumination.

Global illumination

In Figure 4.19 global illumination has been computed. The image is much brighterand the ceiling is illuminated. 200000 photons were used in the global photon map

and 100 photons in the estimate. The caustic photon map parameters are the same.

Photon tracing took 4 seconds. Rendering time was 66 seconds.

The radiance estimate from the global photon map

Finally in Figure 4.20 we have visualized the radiance estimates from the global

photon map directly. We have shown images with 100 and 500 photons in the

estimate. Notice how the illumination becomes softer and more pleasing with more

photons, but also more blurry and with more false color bleeding at the edges. The

edge problem can be solved partially by using an ellipsoid or disc to locate the

photons (see section 4.3) — with 500 photons in the estimate and the ellipsoid

search activated we get the image in Figure 4.21 These images took 30–35 seconds

to render. Notice how the quality of the direct visualization gives a reasonable

82



Figure 4.20: Global photon map radiance estimates visualized directly using 100 pho-

tons (left) and 500 photons (right) in the radiance estimate.

Figure 4.21: Global photon map radiance estimates visualized directly using 500

photons and a disc to locate the photons. Notice the reduced false color bleeding at

the edges.

estimate of the overall illumination in the scene. This is the information we benefit

from in the full rendering step since we do not have to sample the incoming light

recursively.

83



Figure 4.22: Fast visualization of the radiance estimate based on 50 photons and a

global photon map with just 200 photons. Rendering time was 4 seconds.

Fast global illumination estimate

For fast visualization of global illumination one can use very few photons in theglobal photon map. In Figure 4.22 we have visualized the radiance estimate from a

global photon map with just 200 photons! We used up to 50 photons in the radiance

estimate. The illumination is very blurry and as a consequence the shadows and the

caustics are missing, but the overall illumination is approximately correct, and this

visualization is representative of the final rendering as shown in Figure 4.19. Pho-

ton tracing took 0.03 seconds and the rendering time for the image was 4 seconds.

This is almost as fast as the simple ray tracing version, and the main reason is that

we only used ray tracing to compute the first intersection and the mirror reflections

and transmissions. The global photon map was used to estimate both indirect and

direct light.

84



Figure 4.23: Cornell box with water.

4.5.2 Cornell box with water

In the Cornell box in Figure 4.23 we have inserted a displacement-mapped watersurface. To render this scene we used 500000 photons in both the caustics and the

global photon map, and up to 100 photons in the radiance estimate. We used a

higher number of caustic photons due to the water surface which causes the entire

floor to be illuminated by the photons in the caustics photon map. Also the number

of photons in the global photon map have been increased to account for the more

complex indirect illumination in the scene. The water surface is made of 20000

triangles. The rendering time for the image was 11 minutes.

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Figure 4.24: Fractal Cornell box.

4.5.3 Fractal Cornell box

An example of a more complex scene is shown in Figure 4.24. The walls havebeen replaced with displacement mapped surfaces (generated using a fractal mid-

point subdivision algorithm) and the model contains a little more than 1.6 million

elements. Notice that each wall segment is an instanced copy of the same fractal

surface. With photon maps it is easy to take advantage of instancing and the ge-

ometry does not have to be explicitly represented. We used 200000 photons in the

global photon map and 50000 in the caustics photon map. This is the same number

of photons as in the simple Cornell box and our reasoning for choosing the same

values are that the complexity of the illumination is more or less the same as in

the simple Cornell box. We want to capture the color bleeding from the colored

walls and the indirect illumination of the ceiling. All in all we used the same pa-rameters for the photon map as in the simple Cornell box. We only changed the

parameters for the acceleration structure to handle the larger amount of geometry.

The rendering time for the scene was 14 minutes.

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Figure 4.25: Cornell box variation with 4 light sources.

4.5.4 Cornell box with multiple lights

A simple example of a scene with multiple light sources is the variation of theCornell box scene shown in Figure 4.25. We generated 100000 photons from each

light source and the resulting global photon map has 400000 photons. Other than

that the rendering parameters were the same as for the other Cornell box with 1

light source. The rendering time for this scene was 90 seconds.

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Figure 4.26: Cornell box with a participating medium.

4.5.5 Cornell box with smoke

The Cornell box scene shown in Figure 4.26 is an example of a scene with a uni-form participating medium. To simulate this scene we used 100000 photons in the

global photon map and 150000 photons in the volume photon map. A simple non-

adaptive ray marcher has been implemented so the step size had to be set to a low

value which is extra costly. The rendering time for the scene was 44 minutes.

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Figure 4.27: A cognac glass with a caustic.

4.5.6 Cognac glass

Figure 4.27 shows an example of a caustic from a cognac glass. The glass is anobject of revolution approximated with 12000 triangles. To generate the caustic we

used 200000 photons. The radiance estimates for the caustic were computed using

up to 40 photons. The rendering time for the image was 8 minutes and 10 seconds

— part of this rendering time is due to the ray traced depth of field simulation.

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Figure 4.28: Caustics through a prism with dispersion.

4.5.7 Prism with dispersion

The classic example of dispersion with glass prism is shown in Figure 4.28. Eventhough only three separate wavelengths have been sampled, the color variations

in the caustics are smooth. An accurate color representation would require more

wavelength samples; such an extension to the photon map is easy to implement.

500000 photons were used in the caustics and 80 photons were used in the radiance

estimate. The rendering time for the image was 32 seconds.

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Figure 4.29: Granite bunny next to a marble bunny — both models are rendered using subsurface scattering. The photon map is used to compute multiple scattering inside

the stone material.

4.5.8 Subsurface scattering

A recent addition to the photon map is the simulation of subsurface scattering [?,

?]. For subsurface scattering we use the photon map to compute the effect of

multiple scattering within a given material. This is often very costly to compute

and therefore mostly omitted from approaches dealing with subsurface scattering.

This is unfortunate since multiple scattering leads to very nice and subtle color

bleeding effects inside the material which improves the quality of the rendering.

Figure 4.29 shows a granite bunny next to a marble bunny. Both of these stone

models are rendered using subsurface scattering with 100000 photons used to sim-

ulate multiple scattering. The rendering time for the picture was 21 minutes. This

bunny is the original Stanford bunny and the scene contains 140000 triangles, and

it is rendered with full global illumination and depth of field.

Figure 4.30 shows a bust of Diana the Huntress made of translucent marble. For

this scene the light source was behind the bust to emphasize the effect of subsurface

scattering. Notice the translucency of the hair and the nose region. This image was

rendered in 21 minutes using 200000 photons.

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Figure 4.30: Translucent marble bust illuminated from behind

4.6 Where to get programs with photon mapsPhoton maps are already available on the Internet for downloading. We have col-

lected the following links as of the writing of these notes.

RenderPark (a photorealistic rendering tool) has photon maps (as well as many

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other global illumination algorithms). See

http://www.cs.kuleuven.ac.be/cwis/research/graphics/RENDERPARK/ for moreinformation.

Most commercial renderers now supports photon mapping for global illumina-

tion.

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Chapter 5

Photon mapping for complex

scenes

In this chapter, we present methods for photon emission for complex light sources

and photon scattering for complex surface shaders (including displacement shaders),

show how to precompute radiosity estimates for efficient final gathering, describe

the radiosity atlas method for photon mapping in very large scenes, and finally

discuss the use of importance to guide the storage of photons.

In this chapter we will use two scenes from the Pixar movies Ratatouille and

Monsters, Inc. as examples. However, it should be emphasized that photon map-

ping was not used in the production of those movies; we’re just using these scenes

here as examples of how photon mapping could be used in a movie production

setting with very complex illumination, shaders, and geometry.

5.1 Photon emission from complex light sources

Emitting photons corresponding to simple light sources such as point lights, spot

lights, and directional lights is fairly straightforward. However, movie production

light source shaders are very complex: they can project images like a slide projec-

tor and they can have barn doors, cucoloris (“cookies”), unnatural distance fall-off,

fake shadows, artistic positioning of highlights, etc. [5]. For photon emission cor-

responding to such a light source, we would need to compute a photon emission

probability distribution function that corresponds to the light source shader. This

can be difficult since the programmable fall-off requires evaluation of the shader

not only at different angles but also at different distances.

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However, there is a simple method to create photon emission distributions that

exactly match very general light sources. The method starts by evaluating the lightsource shader on the surface points. This is done by rendering the scene with di-

rect illumination from the light source(s) and storing a point cloud of the direct

illumination. The point cloud contains an illumination point for each surface shad-

ing point (roughly one point per pixel). This is a sampling of the light source

shader exactly at the positions where its values matter (namely at the surfaces in

the scene), and ensures that the photon distribution exactly matches the illumina-

tion — no matter how complex and unpredictable the light source shader is. We

basically treat the light source shader as a “black box” that is characterized only by

its illumination on the surfaces in the scene.

On specular or partially specular surfaces, the illumination has to be computedfor each light source separately, and the illumination at each point must be stored

with a vector indicating the incident light direction. For purely diffuse surfaces,

the illumination from all light sources can simply be added up (and stored without

the light direction vector).

Figure 5.1 shows the direct illumination on the diffuse surfaces of the Rata-

touille kitchen. The illumination is specified with more than 200 light sources,

each with rather complex illumination characteristics. (Rendering this image at

resolution 720×405 takes around 42 minutes on 1 processor — this is an indica-

tion of just how complex the light source shaders are.) The generated point cloud

contains 4.8 million points.

Figure 5.1: Point cloud showing the illumination from over 200 light sources in the

Ratatouille kitchen.

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The next step is to emit photons with probability proportional to the stored

illumination power. As always with photon mapping, we must ensure that all emit-ted (and all stored) photons have the same power. This is necessary in order to

minimize noise in the irradiance estimates.

What follows is a more detailed overview of the photon emission algorithm:

First we compute the total power of all illumination in the direct illumination point

cloud. (The incident power at each point is the point’s irradiance times its area.)

The target power per photon is the total power in the point cloud divided by the

specified number of photons to emit. Then, for each point in the illumination point

cloud, we compute the number of photons to emit for that point as the power of the

point divided by the target power per photon. This will usually result in a fractional

number of photons; we use Russian roulette [4] to round up or down to an integer

number and adjust the power of the emitted photon(s) accordingly. The photon(s)

are emitted by shooting them from a point just above the position of the point

cloud point (jittered within the radius of the point). Since the photons are emitted

probabilistically for each point cloud point, the final number of emitted photons

may end up being slightly different from the specified target number.

(A possible improvement of the method would stratify the random numbers

such that no matter what sequence of pseudorandom numbers is encountered, the

Russian roulette will not result in entire regions where the number of emitted pho-

tons is rounded down and other regions where it is rounded up. Quasi-random

numbers could be used as well.)

The idea of emitting photons from a point cloud of direct illumination values

has also been utilized by Hasan et al. [31]. Their photon emission was from and

to a sparse point cloud, and was used to compute an estimate of the multi-bounce

global illumination in a very coarse point-cloud representation of a scene.

5.2 Photon scattering from complex surfaces

In this section we’ll look at photon scattering from surfaces with complex scatter-

ing characteristics.

For simple surface shaders, it is straightforward to write a corresponding pho-

ton shader. Given an incident photon, the photon shader must stochastically deter-

mine whether the photon should be scattered — and if so, the power and direction

of the scattered photon. However, just as for the photon emission case discussed

above, the surface shaders used in movie production are very, very complex. They

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have hundreds or thousands of parameters, and consists of tens of thousands lines

of code. In the worst case, some shaders might be “legacy code” that very fewpeople, if any, have a complete understanding of. Writing photon shaders that

correspond to such surface shaders would be quite a nightmare.

Instead, we can pry the surface reflection coefficients out of the shader. In the

simplest case, the reflection parameter is just a diffuse color; we can obtain that

color by illuminating the scene with single ambient light source with intensity 1.

Figure 5.2 shows the diffuse colors (diffuse reflection coefficients) in the Rata-

touille kitchen scene. The scene has around 150 different surface shaders and more

than 1000 textures (including a few shadow maps).

(a)

(b)

Figure 5.2: Diffuse colors in kitchen: (a) rendered image; (b) point cloud.

For more complex shaders, there might be coefficients for diffuse reflection,

specular reflection, diffuse transmission, and specular transmission, as well as one

or more values specifying the narrowness of the specular (glossy) scattering. The

reflection coefficients may depend on dozens of textures combined in strange ways.

But no matter how complex the shader is, we assume that the result of it can be

expressed by a few reflection coefficients that are stored in a point cloud.

The stored colors are used as scattering coefficients during the photon scatter-

ing step. They are used to calculate the scattering probability and the power of

scattered photons. The point cloud of scattering coefficients is organized into a

kd-tree for fast lookups. When a photon hits a surface, we do a kd-tree lookup in

the point cloud to determine the coefficients at the hit point. The photon is stored if

appropriate and scattered if appropriate (using the usual Russian roulette strategy

for computing photon scattering probilities and new photon power).

This method also handles displacement-mapped surfaces gracefully. If a sur-

face is displaced, that displacement will be present both in the point positions in

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the stored point cloud of scattering coefficients and in the photon hit positions used

for lookups in the kd-tree.

5.3 The radiosity map

In the standard photon mapping method, when a final gather ray hits a point, the

nearest n photons are looked up in the photon map kd-tree in order to estimate the

irradiance at the hit point.

However, it is more efficient to precompute the irradiance at some or all the

photon positions. Then only the nearest photon needs to be looked up at final

gather ray hit points [11]. The irradiance precomputation is quick, and makes the

rendering 5–7 times faster for typical scenes without degrading the image quality.A further optimization is to store the local diffuse surface color with each photon,

and estimating radiosity instead of irradiance at the photon positions [12]. Es-

timating radiosity takes only a tiny bit longer than estimating irradiance (only a

few extra multiplications per photon), but makes it faster to compute the indirect

illumination — especially if the scene has complex shaders and many textures.

Each radiosity estimate is computed by locating the nearest photons, adding up

their power, dividing by the area they cover, and multiplying by the diffuse surface

color.

The result of this step is a new point cloud with position, normal, radius and

radiosity data. We call this point cloud a radiosity map. (The “radiosity” namehere denotes that the data in the point cloud are radiosities; it does not imply that

they have been computed with a finite-element “radiosity method”.) The point

positions divide the surfaces into a Voronoi diagram with constant radiosity inside

each Voronoi cell. These radiosity values can be visualized directly for a rough

estimate of the global illumination in the scene. Figure 5.3 shows the kitchen scene

rendered with the radiosity map values.

Figure 5.4(a) shows the Ratatouille kitchen rendered with only direct illumi-

nation. Figure 5.4(b) shows the same scene with added indirect illumination com-

puted with final gathering using the radiosity map shown above.

Figure 5.5 shows another example of photon map global illumination. This

scene has simple base geometry but texture maps and displaced surfaces: the floor,

left and right wall are textured, and the left and back wall and front teapot are

displacement mapped. Figure 5.5(a) shows the direct illumination in the scene.

Figure 5.5(b) clearly shows the tinting typically associated with color bleeding.

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Figure 5.3: Radiosity map for kitchen.

(a) (b)

Figure 5.4: Ratatouille kitchen: (a) direct illumination; (b) global illumination.


Furthermore, notice that all illumination on the ceiling is indirect light.

The same precomputation method can also be used for precomputing radiances

in a volume photon map. This can speed up ray marching through an isotropically

reflecting participating medium quite significantly.

5.4 The radiosity atlas for large scenes

Here we extend the photon mapping method to enable it to handle scenes with a lot

of geometry. A limitation of the original photon map method, and also the radiosity

map method presented in the previous section, is that they assume that the entire

photon map is stored in memory. If the scene is so complex (contains so much

geometry) that the number of photons required to represent the illumination in it

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(a) (b)

Figure 5.5: A box with textures and displacements: (a) direct illumination;

(b) global illumination.

exceeds the memory capacity of the computer, we need to divide the photon map

into “chunks” that can be read, processed, and cached independently.

The scene used as example in this section is from the Pixar movie Monsters,

Inc.: a city block of Monstropolis with many individually modeled buildings, trees,

cars, etc. The scene geometry consists of 36,000 high-level primitives, mostly

NURBS patches and subdivision surfaces. The shaders have been simplified for

these tests, and only a single light source is used. Figure 5.6 shows the Monstropo-

lis city block with direct illumination from a single point light source and purely

diffuse reflection. Large parts of the scene are completely black since no directlight reaches them.

Figure 5.6: Direct illumination in the Monstropolis scene.

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5.4.1 Photon emission and photon tracing

In the first step, we divide the objects into groups. For the Monstropolis scene we

use the groups created during modeling, so e.g. each house and each car is a group

of objects. Each group gets a separate photon map. Figure 5.7 shows two of the

photon maps for the Monstropolis scene. Both the photon powers and the diffuse

surface colors are shown. The photon map for the car contains 76,000 photons,

while the photon map for the building contains 2.2 million photons. The photons

powers get a green tint when refracted through the windshield. Also notice the red

and blue diffusely reflected photons on the building.

Figure 5.7: Photon maps for the car and building. The two left images show the

photon powers, the two right images show the diffuse surface colors.

We call a collection of photon maps a photon atlas. Figure 5.8 shows the

photon atlas for the Monstropolis scene. For clarity, the figure only shows 0.1% of

the 52 million photons in the photon atlas.

5.4.2 Radiosity estimation

As in section 5.3, we sort the photons in a photon map into a kd-tree and pre-

compute the radiosity at each photon location. We do this independently for each

photon map. Figure 5.9 shows two of the resulting radiosity maps for the Mon-

stropolis scene.

5.4.3 Generating radiosity brick maps

The next step is to compute a brick map representation of each of the radiosity

point clouds.

A brick map is a tiled 3D MIP map data structure [15]. It is a useful data

structure for representing general 3D textures — both textures on surfaces and in

volumes. (Brick maps can also be rendered and ray traced directly as geometric

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Figure 5.8: Coarse photon atlas (collection of photon maps) for the Monstropolis

scene. The photon powers are shown.

Figure 5.9: Radiosity maps for the car and building.

primitives [14].) A brick consists of 8×8×8 voxel positions. The coarsest repre-

sentation in a brick map consists of a single brick. The next level consists of (up

to) 2×2×2 bricks. The next level consists of (up to) 4×4×4 bricks, etc. The ad-

vantage of a brick map representation over a point cloud is that the brick map is

hierarchical and tiled, and that the bricks can be cached efficiently.

For efficient global illumination, we use brick maps to represent the radiosity

data in each radiosity map. Two of the radiosity brick maps for the Monstropolis

scene are shown in figure 5.10. We call the collection of radiosity brick maps a

radiosity brick atlas or simply a radiosity atlas.

Figure 5.11 shows the entire Monstropolis scene rendered with radiosity di-

rectly from the radiosity atlas. This image gives a rough indication of the global

illumination in the scene, but it is far too noisy and blurry to be used in a movie.

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level 0 level 1 level 2 level 3 level 4

Figure 5.10: Radiosity brick maps for the car and building.

Figure 5.11: The Monstropolis scene rendered with radiosity from the radiosity

atlas.

5.4.4 Rendering

When a final gather ray hits a surface point, the ray differential determines the

appropriate brick map level to look up in: we choose the brick map level where the

brick voxels are approximately the size of the ray beam cross-section (similar to

how we select 2D texture levels and tessellation levels).

Figure 5.12 shows a final gather rendering of the scene. At the final gather

ray hit points, the radiosity is determined from the radiosity atlas shown in fig-

ure 5.11. (Rendering this image took 5.7 hours on a 2 GHz Apple G5. It required

the shooting of 413 million final gather rays and 3.8 million shadow rays. There

were 3.4 billion brick cache lookups with a hit rate of 99.9%.)

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Figure 5.12: Monstropolis city block with global illumination: direct illumination

from the sun and sky, and indirect illumination computed using final gathering and

the radiosity atlas. (Copyright c Disney/Pixar.)

5.5 Importance for photon tracing

This section describes the use of importance to concentrate the photons in those

parts of the scene where they contribute most to the final image.

5.5.1 Importance

The first use of importance was for computer simulations of neutron transport forthe development of the hydrogen bomb (right after World War II). Fortunately im-

portance can also be used for more peaceful purposes. Smits et al. [69] formally

introduced importance to computer graphics. They defined importance as the ad-

joint of radiosity that has a source term at the viewpoint, and used importance to

reduce the number of links in a hierarchical radiosity solution. More generally, im-

portance can be defined as the adjoint of any representation of light with the source

term at the viewpoint or at the directly visible points. An overview of importance

and its use in computer graphics can be found in Christensen [13]. The savings

obtained using importance can be arbitrarily high: just choose a sufficiently large

scene with a sufficiently small visible part.

Figure 5.13(a) shows a modest example scene, four rooms with closed doors

between them. Figure 5.13(b) is the image we are interested in computing, a close-

up of the tabletop in the upper right room. The red pyramid in figure 5.13(a)

indicates the viewpoint and viewing frustum for figure 5.13(b).

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(a) (b)

Figure 5.13: Orange interior: (a) entire scene seen from above; (b) seen from the

intended viewpoint.

5.5.2 Importance emission and estimation

Importance can be used to focus the photon storage in those parts of the scene

where the photons will contribute most to the final image.

The first step is to emit importance particles (“importons”) [54, 40, 72] from

the intended viewpoint in directions within the viewing frustum. These particles

are traced around the scene as if they were photons and stored every time they

hit a diffuse surface. The stored particles for the example scene are shown in

figure 5.14(a). (The emitted importance particles are white, but they change color

at each bounce according to the color of the surfaces they hit. In this scene mostparticles turn orange.) After the importance particle tracing, the importance is

estimated at each importance particle location using the local density of importance

particles. These importance estimates are stored with the importance particles; the

estimates are shown in figure 5.14(b). Note that no importance reaches adjacent

rooms since the doors are closed; this reflects the fact that no light from adjacent

rooms reaches the visible parts of the scene.

These precomputed importances make it much faster to determine the impor-

tance at various locations during photon tracing, as described in the following.

5.5.3 Photon tracing

In the photon tracing phase, the importance estimates are used to determine photon

storage probabilities. At locations with low importance, we use Russian roulette

to decide whether to store the photon or not; if the photon is stored, its power is

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(a) (b)

Figure 5.14: Importance in interior scene, seen from above: (a) 100,000 impor-

tance particles; (b) importance estimates.

increased to compensate for the low storage probability. Compare figures 5.15(a–

c) with 5.15(d–f): Importance was not used in figures 5.15(a–c), so most photons

are stored in bright regions, no matter how unimportant those regions are. In con-

trast, figures 5.15(d–f) show the gain from using importance — most photons are

stored in areas that are either directly visible from the intended viewpoint, or are

significantly influencing the illumination there. Due to the higher concentration of

photons, the illumination is approximated much more accurately in figure 5.15(f)

than in figure 5.15(c). The most visible difference is the sharper shadows.

(a) (b) (c)

(d) (e) (f)

Figure 5.15: Light in interior scene: (a) 500,000 photons stored without impor-

tance; (b)-(c) radiance estimates based on the photons in (a); (d) 500,000 photonsstored using importance; (e)-(f) radiance estimates based on the photons in (d).

In this example, importance was only used to determine photon storage. It is

also possible to use importance to guide photon emission and scattering [54, 12],

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but that is beyond the scope of this course note.

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Bibliography

[1] Oliver Abert, Markus Geimer, and Stefan Muller. Direct and fast ray tracing

of NURBS surfaces. In Proceedings of the IEEE Symposium on Interactive

Ray Tracing 2006 , pages 161–168. IEEE, 2006.

[2] Anthony A. Apodaca and Larry Gritz. Advanced RenderMan: Creating CGI

for Motion Pictures. Morgan Kaufmann Publishers, 2000.

[3] Arthur Appel. Some techniques for shading machine renderings of solids.

In Proceedings of the AFIPS Spring Joint Computer Conference, volume 32,

pages 37–45, 1968.

[4] James R. Arvo and David B. Kirk. Particle transport and image synthesis.

Computer Graphics (Proceedings of SIGGRAPH ’90), 24(4):63–66, 1990.

[5] Ronen Barzel. Lighting controls for computer cinematography. Journal of

Graphics Tools, 2(1):1–20, 1997.

[6] James F. Blinn. Models of light reflection for computer synthesized pictures.


[7] James F. Blinn and Martin E. Newell. Texture and reflection in computer

generated images. Communications of the ACM , 19(10):542–547, 1976.

[8] Phong Bui Tuong. Illumination for computer generated pictures. Communi-

cations of the ACM , 18(3):311–317, 1975.

[9] Edwin E. Catmull. A Subdivision Algorithm for Computer Display of Curved

Surfaces. PhD thesis, University of Utah, Salt Lake City, 1974.

[10] Edwin E. Catmull and James H. Clark. Recursively generated B-spline sur-

faces on arbitrary topological meshes. Computer-Aided Design, 10(6):350–

355, 1978.

109



[11] Per H. Christensen. Faster photon map global illumination. Journal of Graph-

ics Tools, 4(3):1–10, 1999.

[12] Per H. Christensen. Photon mapping tricks. In SIGGRAPH 2002 Course Note

#43: A Practical Guide to Global Illumination using Photon Mapping, pages

93–121, 2002.

[13] Per H. Christensen. Adjoints and importance in rendering: an overview. IEEE

Transactions on Visualization and Computer Graphics, 9(3):329–340, 2003.

[14] Per H. Christensen. Point clouds and brick maps for movie production. In

Markus Gross and Hanspeter Pfister, editors, Point-Based Graphics, chapter

8.4. Morgan Kaufmann Publishers, 2007.

[15] Per H. Christensen and Dana Batali. An irradiance atlas for global illumina-

tion in complex production scenes. In Rendering Techniques 2004 (Proceed-

ings of the Eurographics Symposium on Rendering 2004), pages 133–141.

Eurographics, 2004.

[16] Per H. Christensen, Julian Fong, David M. Laur, and Dana Batali. Ray tracing

for the movie ‘Cars’. In Proceedings of the IEEE Symposium on Interactive

Ray Tracing 2006 , pages 1–6. IEEE, 2006.

[17] Per H. Christensen, David M. Laur, Julian Fong, Wayne L. Wooten, and Dana

Batali. Ray differentials and multiresolution geometry caching for distribu-

tion ray tracing in complex scenes. Computer Graphics Forum (Proceedings

of Eurographics 2003), 22(3):543–552, 2003.

[18] Robert L. Cook, Loren Carpenter, and Edwin Catmull. The Reyes image

rendering architecture. Computer Graphics (Proceedings of SIGGRAPH ’87),

21(4):95–102, 1987.

[19] Robert L. Cook, Thomas Porter, and Loren Carpenter. Distributed ray tracing.


[20] Tony D. DeRose, Michael Kass, and Tien Truong. Subdivision surfaces in

character animation. Computer Graphics (Proceedings of SIGGRAPH ’98),pages 85–94, 1998.

[21] Daniel Doo and Malcolm A. Sabin. Behaviour of recursive division surfaces

near extraordinary points. Computer-Aided Design, 10(6):356–360, 1978.

110



[22] Albrecht Durer. Treatise on measurement with compasses and straightedge

(Underweysung der Messung mit dem Zirkel und Richtscheyt). Nuremberg,1525.

[23] Gerald Farin. Curves and Surfaces for CAGD: A Practical Guide. Academic

Press, 3rd edition, 1993.

[24] James D. Foley, Andries van Dam, Steven K. Feiner, and John F. Hughes.

Computer Graphics: Principles and Practice. Addison-Wesley Publishing

Company, 2nd edition, 1990.

[25] Andrew S. Glassner. An Introduction to Ray Tracing. Academic Press, 1989.

[26] Andrew S. Glassner. Principles of Digital Image Synthesis. Morgan Kauf-mann Publishers, 1995.

[27] Jeffrey Goldsmith and John Salmon. Automatic creation of object hierar-

chies for ray tracing. IEEE Computer Graphics and Applications, 7(5):14–20,

1987.

[28] Ned Greene. Environment mapping and other applications of world projec-

tions. IEEE Computer Graphics and Applications, 6(11):21–29, 1986.

[29] Eric Haines. Ray Tracing News. 1987–present. (Web page: www.acm.org/-

tog/resources/RTNews/html).

[30] Vlastimil Havran. Heuristic Ray Shooting Algorithms. PhD thesis, Czech

Technical University, Prague, 2001.

[31] Milos Hasan, Fabio Pellacini, and Kavita Bala. Direct-to-indirect transfer

for cinematic relighting. ACM Transactions on Graphics (Proceedings of

SIGGRAPH 2006), 25(3):1089–1097, 2006.

[32] Paul S. Heckbert. Survey of texture mapping. IEEE Computer Graphics and

Applications, 6(11):56–67, 1986.

[33] Abu Sad al-Ala ibn Sahl. On Burning Mirrors and Lenses. Baghdad, 984.(Translated by Roshdi Rashed, 1990).

[34] Homan Igehy. Tracing ray differentials. Computer Graphics (Proceedings of

SIGGRAPH ’99), pages 179–186, 1999.

111



[35] Henrik Wann Jensen, James Arvo, Philip Dutre, Alexander Keller, Art Oven,

Matt Pharr, and Peter Shirley. SIGGRAPH 2003 Course Note #44: MonteCarlo Ray Tracing. 2003.

[36] James T. Kajiya. Ray tracing parametric patches. Computer Graphics (Pro-

ceedings of SIGGRAPH ’82), 16(3):245–254, 1982.

[37] James T. Kajiya. The rendering equation. Computer Graphics (Proceedings

of SIGGRAPH ’86), 20(4):143–150, 1986.

[38] Toshiaki Kato. The Kilauea massively parallel ray tracer. In Alan Chalmers,

Timothy Davis, and Erik Reinhard, editors, Practical Parallel Rendering,

chapter 8. A K Peters, 2002.

[39] Timothy L. Kay and James T. Kajiya. Ray tracing complex scenes. Computer

Graphics (Proceedings of SIGGRAPH ’86), 20(4):269–278, 1986.

[40] Alexander Keller and Ingo Wald. Efficient importance sampling techniques

for the photon map. In Proceedings of the 5th Fall Workshop on Vision,

Modeling, and Visualization, pages 271–279, 2000.

[41] Leif Kobbelt, Katja Daubert, and Hans-Peter Seidel. Ray tracing of subdi-

vision surfaces. In Rendering Techniques ’98 (Proceedings of the 9th Euro-

graphics Workshop on Rendering), pages 69–80, 1998.

[42] Craig Kolb, Pat Hanrahan, and Don Mitchell. A realistic camera model for

computer graphics. Computer Graphics (Proceedings of SIGGRAPH ’95),

pages 317–324, 1995.

[43] Ares Lagae and Philip Dutre. An efficient ray-quadrilateral intersection test.

Journal of Graphics Tools, 10(4):23–32, 2005.

[44] Johann H. Lambert. Photometry: or on the Measure and Gradations of Light,

Colors, and Shade. 1760. (Translated from the Latin by David L. DiLaura,

2001).

[45] Hayden Landis. Production-ready global illumination. In SIGGRAPH 2002Course Note #16: RenderMan in Production, pages 87–102, 2002.

[46] Charles Loop. Smooth subdivision surfaces based on triangles. Master’s

thesis, University of Utah, Salt Lake City, 1987.

112



[47] William Martin, Elaine Cohen, Russel Fish, and Peter Shirley. Practical ray

tracing of trimmed NURBS surfaces. Journal of Graphics Tools, 5(1):27–52,2000.

[48] Tomas Moller and Ben Trumbore. Fast, minimum storage ray triangle inter-

section. Journal of Graphics Tools, 2(1):21–28, 1997.

[49] Michael J. Muuss. Rt and remrt — shared memory parallel and network dis-

tributed ray-tracing programs. In USENIX: Proceedings of the Fourth Com-

puter Graphics Workshop, 1987.

[50] Isaac Newton. Opticks: A Treatise on the Reflections, Refractions, Inflections

and Colours of Light . London, 1704.

[51] Michael Oren and Shree K. Nayar. Generalization of Lambert’s reflectance

model. Computer Graphics (Proceedings of SIGGRAPH ’94), pages 239–

246, 1994.

[52] Steven Parker, William Martin, Peter-Pike J. Sloan, Peter Shirley, Brian

Smits, and Charles Hansen. Interactive ray tracing. In Symposium on In-

teractive 3D Graphics, pages 119–126, 1999.

[53] Darwyn Peachey. Texture on demand. Technical Report #217, Pixar, 1990.

(Available at graphics.pixar.com).

[54] Ingmar Peter and Georg Pietrek. Importance driven construction of photon

maps. In Rendering Techniques ’98 (Proceedings of the 9th Eurographics

Workshop on Rendering), pages 269–280, 1998.

[55] Matt Pharr and Pat Hanrahan. Geometry caching for ray-tracing displacement

maps. In Rendering Techniques ’96 (Proceedings of the 7th Eurographics

Workshop on Rendering), pages 31–40, 1996.

[56] Matt Pharr and Greg Humphreys. Physically Based Rendering: From Theory

to Implementation. Morgan Kaufmann Publishers, 2004.

[57] Matt Pharr, Craig Kolb, Reid Gershbein, and Pat Hanrahan. Rendering com-plex scenes with memory-coherent ray tracing. Computer Graphics (Pro-

ceedings of SIGGRAPH ’97), pages 101–108, 1997.

[58] Les Piegl and Wayne Tiller. The NURBS Book . Springer-Verlag, 1997.

113



[59] William T. Reeves, David H. Salesin, and Robert L. Cook. Rendering an-

tialiased shadows with depth maps. Computer Graphics (Proceedings of SIG-GRAPH ’87), 21(4):283–291, 1987.

[60] Erik Reinhard, Brian Smits, and Chuck Hansen. Dynamic acceleration struc-

tures for interactive ray tracing. In Rendering Techniques 2000 (Proceedings

of the 11th Eurographics Workshop on Rendering), pages 299–306, 2000.

[61] Christophe Schlick. A customizable reflectance model for everyday render-

ing. In Proceedings of the 4th Eurographics Workshop on Rendering, pages

73–83, 1993.

[62] Andrei Sherstyuk. Fast ray tracing of implicit surfaces. Computer Graphics

Forum, 18(2):139–147, 1999.

[63] Peter Shirley. Fundamentals of Computer Graphics. A K Peters, 2002.

[64] Peter Shirley and R. Keith Morley. Realistic Ray Tracing. A K Peters, 2nd

edition, 2005.

[65] Peter Shirley, Philipp Slusallek, Ingo Wald, et al. SIGGRAPH 2006 Course

Note #4: State of the Art in Interactive Ray Tracing. 2006.

[66] Peter Shirley, Changyaw Wang, and Kurt Zimmerman. Monte Carlo tech-

niques for direct lighting calculations. ACM Transactions on Graphics,15(1):1–36, 1996.

[67] Brian Smits. Efficiency issues for ray tracing. Journal of Graphics Tools,

3(2):1–14, 1998.

[68] Brian Smits, Peter Shirley, and Michael M. Stark. Direct ray tracing of dis-

placement mapped triangles. In Rendering Techniques 2000 (Proceedings of

the 11th Eurographics Workshop on Rendering), pages 307–318, 2000.

[69] Brian E. Smits, James R. Arvo, and David H. Salesin. An importance-driven

radiosity algorithm. Computer Graphics (Proceedings of SIGGRAPH ’92),26(2):273–282, 1992.

[70] Ian Stephenson, editor. Production Rendering: Design and Implementation.

Springer-Verlag, 2005.

114



[71] Gordon Stoll, William R. Mark, Peter Djeu, Rui Wang, and Ikrima Elhassan.

Razor: an architecture for dynamic multiresolution ray tracing. TechnicalReport TR-06-21, University of Texas at Austin, 2006. (Updated version to

appear in ACM Transactions on Graphics).

[72] Frank Suykens and Yves D. Willems. Density control for photon maps. In

Rendering Techniques 2000 (Proceedings of the 11th Eurographics Workshop

on Rendering), pages 11–22, 2000.

[73] Frank Suykens and Yves D. Willems. Path differentials and applications. In

Rendering Techniques 2001 (Proceedings of the 12th Eurographics Workshop

on Rendering), pages 257–268, 2001.

[74] Steve Upstill. The RenderMan Companion. Addison Wesley Publishers,

1990.

[75] Ingo Wald, Andreas Dietrich, and Philipp Slusallek. An interactive out-of-

core rendering framework for visualizing massively complex models. In

Rendering Techniques 2004 (Proceedings of the Eurographics Symposium on

Rendering 2004), pages 81–92, 2004.

[76] Ingo Wald, Thiago Ize, Andrew Kensler, Aaron Knoll, and Steven G. Parker.

Ray tracing animated scenes using coherent grid traversal. ACM Transactions

on Graphics (Proceedings of SIGGRAPH 2006), 25(3):485–493, 2006.

[77] Ingo Wald and Steven G. Parker, editors. Proceedings of the IEEE Symposium

on Interactive Ray Tracing 2006 . IEEE, 2006. (Web page: www.sci.utah.edu-

/RT06).

[78] Ingo Wald, Philipp Slusallek, Carsten Benthin, and Michael Wagner. Inter-

active distributed ray tracing of highly complex models. In Rendering Tech-

niques 2001 (Proceedings of the 12th Eurographics Workshop on Rendering),

pages 277–288, 2001.

[79] Ingo Wald, Philipp Slusallek, Carsten Benthin, and Michael Wagner. Inter-

active rendering with coherent raytracing. Computer Graphics Forum (Pro-ceedings of Eurographics 2001), 20(3):153–164, 2001.

[80] Gregory J. Ward. Adaptive shadow testing for ray tracing. In Proceedings of

the 2nd Eurographics Workshop on Rendering, pages 11–20, 1991.

115



[81] Gregory J. Ward. Measuring and modeling anisotropic reflection. Computer

Graphics (Proceedings of SIGGRAPH ’92), 26(2):265–272, 1992.

[82] Gregory J. Ward, Francis M. Rubinstein, and Robert D. Clear. A ray trac-

ing solution for diffuse interreflection. Computer Graphics (Proceedings of

SIGGRAPH ’88), 22(4):85–92, 1988.

[83] Turner Whitted. An improved illumination model for shaded display. Com-

munications of the ACM , 23(6):343–349, 1980.

[84] Lance Williams. Pyramidal parametrics. Computer Graphics (Proceedings

of SIGGRAPH ’83), 17(3):1–11, 1983.

[85] Sergei Zhukov, Andrei Iones, and Gregorij Kronin. An ambient light illu-mination model. In Rendering Techniques ’98 (Proceedings of the 9th Euro-

graphics Workshop on Rendering), pages 45–55, 1998.

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